`International Journal of Differential EquationsVolume 2011, Article ID 548982, 9 pageshttp://dx.doi.org/10.1155/2011/548982`
Research Article

## Generalized Differential Transform Method to Space-Time Fractional Telegraph Equation

1Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India
2Department of Computer Engineering, Vyas Institute of Higher Education, Jodhpur 342001, Rajasthan, India

Received 28 May 2011; Revised 20 July 2011; Accepted 23 July 2011

Copyright © 2011 Mridula Garg et al. 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

We use generalized differential transform method (GDTM) to derive the solution of space-time fractional telegraph equation in closed form. The space and time fractional derivatives are considered in Caputo sense and the solution is obtained in terms of Mittag-Leffler functions.

#### 1. Introduction

Differential equations of fractional order have been successfully employed for modeling the so called anomalous phenomena during last two decades. As a consequence, there has been an intensive development of the theory of fractional differential equations [14]. Recently, various analytical and numerical methods have been employed to solve linear and nonlinear fractional differential equations. A few to mention are Adomian decomposition method [57], homotopy perturbation method [8, 9], homotopy analysis method [10], variational iteration method [11, 12], matrix method [13], and differential transform method [1416]. The differential transform method was proposed by Zhou [17] to solve linear and nonlinear initial value problems in electric circuit analysis. This method constructs an analytical solution in the form of a polynomial. It is different from the traditional higher order Taylor series method, which requires symbolic computation of the necessary derivatives of the data functions and takes long time in computation, whereas the differential transform is an iterative procedure for obtaining analytic Taylor series solution. The method is further developed by Momani, Odibat, and Erturk in their papers [1416] for solving two-dimensional linear and nonlinear partial differential equations of fractional order. Recently, Biazar and Eslami [18] applied differential transform method to solve systems of Volterra integral equations of the first kind, El-Said et al. [19] developed extended Weierstrass transformation method for nonlinear evolution equations, and Keskin and Oturanc [20] developed the reduced differential transform method to solve fractional partial differential equations.

In the present paper, we apply the method of generalized differential transform to solve space-time fractional telegraph equation. The classical telegraph equation is a partial differential equation with constant coefficients given by [21] where , and are constants. This equation is used in modeling reaction diffusion and signal analysis for propagation of electrical signals in a cable of transmission line [21, 22]. Both current I and voltage V satisfy an equation of the form (1.1). This equation also arises in the propagation of pressure waves in the study of pulsatile blood flow in arteries and in one-dimensional random motion of bugs along a hedge. Compared to the heat equation, the telegraph equation is found to be a superior model for describing certain fluid flow problems involving suspensions [23]. This equation is used in modeling reaction diffusion and signal analysis for transmission and propagation of electrical signals.

The classical telegraph equation and space or time fractional telegraph equations have been studied by a number of researchers namely Biazar et al. [24], Cascaval et al. [25], Kaya [26], Momani [5], Odibat and Momani [27], Sevimlican [12], and Yıldırim [9]. Orsingher and Zhao [28] have shown that the law of the iterated Brownian motion and the telegraph processes with Brownian time are governed by time-fractional telegraph equations. Orsingher and Beghin [29] presented that the transition function of a symmetric process with discontinuous trajectories satisfies the space-fractional telegraph equation. Several techniques such as transform method, Adomian decomposition method, juxtaposition of transforms, generalized differential transform method, variational iteration method, and homotopy perturbation method have been used to solve space or time fractional telegraph equation.

In the present paper, we make an attempt to solve homogeneous and nonhomogeneous space-time fractional telegraph equation by means of generalized differential transform method.

#### 2. Preliminaries

Definition 2.1. Caputo fractional derivative of order is defined as [30]:

Definition 2.2. The Mittag-Leffler function which is a generalization of exponential function is defined as [31]: A further generalization of (2.2) is given in the form [32] For reduces to .

Definition 2.3. Generalized two-dimensional differential transform [1416] is as given below: Consider a function of two variables and suppose that it can be represented as a product of two single-variable functions, that is, . If function is analytic and differentiated continuously with respect to and in the domain of interest, then the generalized two-dimensional differential transform of the function is given by where , ( times), is defined by (2.1) and is the transformed function.

The generalized differential transform inverse of is given by

Some basic properties of the generalized two-dimensional differential transform are as given below.

Let , and be generalized two-dimensional differential transform of the functions , , and , respectively, then(a)if , then ,(b)if , is constant, then ,(c)if where , then ,(d)if where , , then .

#### 3. Solution of Space-Time Fractional Telegraph Equations by Generalized Two-Dimensional Differential Transform Method

In this section, we consider space-time fractional telegraph equations in the following form: where , , , , , , ( times), ( times), , are Caputo fractional derivatives defined by (2.1), , , and are constants, is given function.

Particularly for , , , , , space-time fractional telegraph (3.1) reduces to classical telegraph (1.1).

To give a clear overview of the methodology, we have selected three illustrative examples, the first is a homogeneous space-time fractional telegraph equation with conditions involving ordinary derivative with respect to space, the second is a homogeneous space-time fractional telegraph equation with conditions involving fractional derivative with respect to space, and the third is a nonhomogeneous space-time fractional telegraph equation with conditions involving fractional derivative with respect to space.

Example 3.1. Consider the following homogeneous space-time fractional telegraph equation: where , , , , , ( times), ( times), , are Caputo fractional derivatives defined by (2.1), is odd and Applying generalized two-dimensional differential transform (2.4) with , , to both sides of (3.2) and (3.3) and using properties (c) and (d), we obtain Utilizing recurrence relation (3.4), the transformed conditions (3.5) and the condition is odd, we can easily obtain, for Now, from (2.5), we have Using the values of from (3.5)–(3.8) in (3.9), the exact solution of space-time fractional telegraph (3.2) is obtained as which is same as obtained by Garg and Sharma [33] using Adomian decomposition method.

Setting , , the space-time fractional telegraph (3.2) reduces to space fractional telegraph equation and the solution is same as obtained by Momani [5], Odibat and Momani [27], and Yıldırim [9] using Adomian decomposition method, generalized differential transform method, and homotopy perturbation method, respectively.

Further, setting , it reduces to classical telegraph equation and the solution is same as obtained by Kaya [26] using Adomian decomposition method.

Example 3.2. Consider the following homogeneous space-time fractional telegraph equation: where , , , , , , ( times), ( times), , are Caputo fractional derivatives defined by (2.1), is odd and Applying generalized two-dimensional differential transform (2.4) with to both sides of (3.11), (3.12) and using properties (c) and (d) we obtain Utilizing the recurrence relation (3.13), the transformed conditions (3.14) and the condition is odd, we obtain Now, from (2.5), we have Using the values of from (3.15) in (3.16), the exact solution of homogeneous space-time fractional telegraph (3.11) is obtained as

Remark 3.3. (1) Setting , , , (3.11) reduces to space fractional telegraph equation with solution
(2) Setting , (3.11) reduces to time fractional telegraph equation: with solution
(3) Setting , , , , (3.11) reduces to classical telegraph equation: with solution which is same as obtained by Kaya [26] using Adomian decomposition method.

Example 3.4. Consider the following non-homogeneous space-time fractional telegraph equation: where , , , , , , ( times), ( times), , are Caputo fractional derivatives defined by (2.1), and are even and Applying generalized two-dimensional differential transform (2.4) with to both sides of (3.24), (3.25), and using properties (c) and (d) we obtain Utilizing the recurrence relation (3.26) and the transformed conditions (3.27), we obtain Now from (2.5), we have Using the values of from (3.28) in (3.29), the exact solution of non-homogeneous space-time fractional telegraph (3.24) is obtained as

Remark 3.5. (1) Setting , , , (3.24) reduces to non-homogeneous space fractional telegraph equation: with solution
(2) Setting , (3.24) reduces to non-homogeneous time fractional telegraph equation: with solution
(3) Setting , , , , (3.24) reduces to non-homogeneous telegraph equation: with solution

#### Acknowledgments

The support provided by the Council of Scientific and Industrial Research through a senior research fellowship to one of the authors, P. Manohar, is gratefully acknowledged. The authors are thankful to the referees for their useful comments and suggestions.

#### References

1. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
2. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006.
3. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
4. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
5. 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.
6. S. S. Ray and R. K. Bera, “Solution of an extraordinary differential equation by Adomian decomposition method,” Journal of Applied Mathematics, no. 4, pp. 331–338, 2004.
7. S. S. Ray and R. K. Bera, “An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 167, no. 1, pp. 561–571, 2005.
8. Q. Wang, “Homotopy perturbation method for fractional KdV-Burgers equation,” Chaos, Solitons and Fractals, vol. 35, no. 5, pp. 843–850, 2008.
9. Ahmet Yıldırım, “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.
10. H. Jafari, C. Chun, S. Seifi, and M. Saeidy, “Analytical solution for nonlinear gas dynamic equation by homotopy analysis method,” Applications and Applied Mathematics, vol. 4, no. 1, pp. 149–154, 2009.
11. J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998.
12. 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.
13. M. Garg and P. Manohar, “Numerical solution of fractional diffusion-wave equation with two space variables by matrix method,” Fractional Calculus & Applied Analysis, vol. 13, no. 2, pp. 191–207, 2010.
14. S. Momani, Z. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation,” Physics Letters. A, vol. 370, no. 5-6, pp. 379–387, 2007.
15. Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008.
16. Z. Odibat, S. Momani, and V. S. Erturk, “Generalized differential transform method: application to differential equations of fractional order,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 467–477, 2008.
17. J. K. Zhou, Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.
18. J. Biazar and M. Eslami, “Differential transform method for systems of Volterra integral equations of the first kind,” Nonlinear Science Letters A, vol. 1, pp. 173–181, 2010.
19. A. El-Said, M. El-Wakil, M. Essam Abulwafa, and A. Mohammed, “Extended weierstrass transformation method for nonlinear evolution equations,” Nonlinear Science Letters A, vol. 1, 2010.
20. Y. Keskin and G. Oturanc, “The reduced differential transform method: a new approach to fractional partial differential equations,” Nonlinear Science Letters A, vol. 1, pp. 207–217, 2010.
21. L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäauser, Boston, Mass, USA, 1997.
22. A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus, London, UK, 1993.
23. E. C. Eckstein, J. A. Goldstein, and M. Leggas, “The mathematics of suspensions: Kac walks and asymptotic analyticity,” in Proceedings of the 4th Mississippi State Conference on Difference Equations and Computational Simulations, vol. 3, pp. 39–50.
24. J. Biazar, H. Ebrahimi, and Z. Ayati, “An approximation to the solution of telegraph equation by variational iteration method,” Numerical Methods for Partial Differential Equations, vol. 25, no. 4, pp. 797–801, 2009.
25. Radu C. Cascaval, E. C. Eckstein, L. Frota, and J. A. Goldstein, “Fractional telegraph equations,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 145–159, 2002.
26. D. Kaya, “A new approach to the telegraph equation: an application of the decomposition method,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 28, no. 1, pp. 51–57, 2000.
27. Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008.
28. E. Orsingher and X. Zhao, “The space-fractional telegraph equation and the related fractional telegraph process,” Chinese Annals of Mathematics. Series B, vol. 24, no. 1, pp. 45–56, 2003.
29. 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.
30. M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, Italy, 1969.
31. G. M. Mittag-Leffler, “Sur la nouvelle fonction ${E}_{\alpha }\left(x\right)$,” Comptes rendus de l' Académie des Sciences Paris, no. 137, pp. 554–558, 1903.
32. A. Wiman, “Über den fundamentalsatz in der teorie der funktionen ${E}_{\alpha }\left(x\right)$,” Acta Mathematica, vol. 29, no. 1, pp. 191–201, 1905.
33. 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.