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International Journal of Differential Equations
Volume 2011 (2011), Article ID 548982, 9 pages
http://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

Academic Editor: Kanishka Perera

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]𝑢𝑡𝑡𝑐2𝑢𝑥𝑥+𝑎𝑢𝑡+𝑏𝑢=0,(1.1) where 𝑎, 𝑏 and 𝑐 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 𝛼 is defined as [30]: 𝐷𝛼𝑎1𝑓(𝑥)=Γ(𝑚𝛼)𝑥𝑎𝑓(𝑚)(𝜉)(𝑥𝜉)𝛼𝑚+1𝑑𝜉,(𝑚1<𝛼𝑚),𝑚.(2.1)

Definition 2.2. The Mittag-Leffler function which is a generalization of exponential function is defined as [31]: 𝐸𝛼(𝑧)=𝑛=0𝑧𝑛(Γ(𝛼𝑛+1)𝛼𝐶,𝑅(𝛼)>0).(2.2) A further generalization of (2.2) is given in the form [32] 𝐸𝛼,𝛽(𝑧)=𝑛=0𝑧𝑛Γ(𝛼𝑛+𝛽);(𝛼,𝛽𝐶,𝑅(𝛼)>0,𝑅(𝛽)>0).(2.3) For 𝛼=1,𝐸𝛼(𝑧) reduces to 𝑒𝑧.

Definition 2.3. Generalized two-dimensional differential transform [1416] is as given below: Consider a function of two variables 𝑢(𝑥,𝑦) and suppose that it can be represented as a product of two single-variable functions, that is, 𝑢(𝑥,𝑦)=𝑓(𝑥)𝑔(𝑦). If function 𝑢(𝑥,𝑦) is analytic and differentiated continuously with respect to 𝑥 and 𝑦 in the domain of interest, then the generalized two-dimensional differential transform of the function 𝑢(𝑥,𝑦) is given by 𝑈𝛼,𝛽1(𝑘,)=𝐷Γ(𝛼𝑘+1)Γ(𝛽+1)𝛼𝑥0𝑘𝐷𝛽𝑦0𝑢(𝑥,𝑦)(𝑥0,𝑦0),(2.4) where 0<𝛼, 𝛽1(𝐷𝛼𝑥0)𝑘=𝐷𝛼𝑥0𝐷𝛼𝑥0𝐷𝛼𝑥0 (𝑘 times), 𝐷𝛼𝑥0 is defined by (2.1) and 𝑈𝛼,𝛽(𝑘,) is the transformed function.

The generalized differential transform inverse of 𝑈𝛼,𝛽(𝑘,) is given by𝑢(𝑥,𝑦)=𝑘=0=0𝑈𝛼,𝛽(𝑘,)𝑥𝑥0𝑘𝛼𝑦𝑦0𝛽.(2.5)

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

Let 𝑈𝛼,𝛽(𝑘,), 𝑉𝛼,𝛽(𝑘,) and 𝑊𝛼,𝛽(𝑘,) be generalized two-dimensional differential transform of the functions 𝑢(𝑥,𝑦), 𝑣(𝑥,𝑦), and 𝑤(𝑥,𝑦), respectively, then(a)if 𝑢(𝑥,𝑦)=𝑣(𝑥,𝑦)±𝑤(𝑥,𝑦), then 𝑈𝛼,𝛽(𝑘,)=𝑉𝛼,𝛽(𝑘,)±𝑊𝛼,𝛽(𝑘,),(b)if 𝑢(𝑥,𝑦)=𝑎𝑣(𝑥,𝑦), 𝑎 is constant, then 𝑈𝛼,𝛽(𝑘,)=𝑎𝑉𝛼,𝛽(𝑘,),(c)if 𝑢(𝑥,𝑦)=𝐷𝛾𝑥0𝑣(𝑥,𝑦) where 𝑚1<𝛾𝑚, 𝑚 then 𝑈𝛼,𝛽(𝑘,)=(Γ(𝛼𝑘+𝛾+1)/Γ(𝛼𝑘+1))𝑉𝛼,𝛽(𝑘+𝛾/𝛼,),(d)if 𝑢(𝑥,𝑦)=𝐷𝜇𝑦0𝑣(𝑥,𝑦) where 𝑛1<𝜇𝑛, 𝑛, then 𝑈𝛼,𝛽(𝑘,)=(Γ(𝛽+𝜇+1)/Γ(𝛽+1))𝑉𝛼,𝛽(𝑘,+𝜇/𝛽).

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:𝑐2𝐷𝑥2𝛼𝑢(𝑥,𝑡)=𝐷𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝑎𝐷𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑏𝑢(𝑥,𝑡)+𝑓(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.1) where 𝛽=1/𝑞, 𝑝,𝑞,𝑟, 1<2𝛼2, 1<𝑝𝛽2, 0<𝑟𝛽1, 𝐷𝑥2𝛼𝐷𝛼𝑥𝐷𝛼𝑥, 𝐷𝑡𝑝𝛽𝐷𝛽𝑡𝐷𝛽𝑡𝐷𝛽𝑡 (𝑝 times), 𝐷𝑡𝑟𝛽𝐷𝛽𝑡𝐷𝛽𝑡𝐷𝛽𝑡 (𝑟 times), 𝐷𝛼𝑥, 𝐷𝛽𝑡 are Caputo fractional derivatives defined by (2.1), 𝑎, 𝑏, and 𝑐 are constants, 𝑓(𝑥,𝑡) is given function.

Particularly for 𝛼=1, 𝑞=1, 𝑝=2, 𝑟=1, 𝑓=0, 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: 𝐷𝑥3/2𝑢(𝑥,𝑡)=𝐷𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝐷𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.2) where 𝛽=1/𝑞, 𝑝,𝑞,𝑟, 1<𝑝𝛽2, 0<𝑟𝛽1, 𝐷𝑥3/2(𝐷𝑥1/2)3, 𝐷𝑡𝑝𝛽𝐷𝛽𝑡𝐷𝛽𝑡𝐷𝛽𝑡 (𝑝 times), 𝐷𝑡𝑟𝛽𝐷𝛽𝑡𝐷𝛽𝑡𝐷𝛽𝑡 (𝑟 times), 𝐷𝛼𝑥, 𝐷𝛽𝑡 are Caputo fractional derivatives defined by (2.1), 𝑝+𝑟 is odd and 𝑢(0,𝑡)=𝐸𝛽𝑡𝛽,𝑢𝑥(0,𝑡)=𝐸𝛽𝑡𝛽.(3.3) Applying generalized two-dimensional differential transform (2.4) with 𝑥0=0=𝑦0, 𝛼=1/2, to both sides of (3.2) and (3.3) and using properties (c) and (d), we obtain 𝑈1/2,𝛽(𝑘+3,)=Γ((𝑘/2)+1)Γ(((𝑘+3)/2)+1)Γ(𝛽(+𝑝)+1)𝑈Γ(𝛽+1)1/2,𝛽+(𝑘,+𝑝)Γ(𝛽(+𝑟)+1)𝑈Γ(𝛽+1)1/2,𝛽(𝑘,+𝑟)+𝑈1/2,𝛽,(𝑘,)(3.4)𝑈1/2,𝛽(0,)=(1)Γ(𝛽+1),𝑈1/2,𝛽𝑈(1,)=0,1/2,𝛽(2,)=(1)Γ(𝛽+1),=0,1,2,.(3.5) Utilizing recurrence relation (3.4), the transformed conditions (3.5) and the condition 𝑝+𝑟 is odd, we can easily obtain, for 𝑙,=0,1,2,𝑈1/2,𝛽(3𝑙,)=(1),𝑈Γ((3/2)𝑙+1)Γ(𝛽+1)(3.6)1/2,𝛽𝑈(3𝑙+1,)=0,(3.7)1/2,𝛽(3𝑙+2,)=(1).Γ((3/2)𝑙+2)Γ(𝛽+1)(3.8) Now, from (2.5), we have 𝑢(𝑥,𝑡)=𝑘=0=0𝑈1/2,𝛽(𝑘,)𝑥𝑘/2𝑡𝛽.(3.9) Using the values of 𝑈1/2,𝛽(𝑘,) from (3.5)–(3.8) in (3.9), the exact solution of space-time fractional telegraph (3.2) is obtained as 𝐸𝑢(𝑥,𝑡)=3/2𝑥3/2+𝑥𝐸3/2,2𝑥3/2𝐸𝛽𝑡𝛽,(3.10) which is same as obtained by Garg and Sharma [33] using Adomian decomposition method.

Setting 𝑝=2, 𝑞=𝑟=1, 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 𝛼=1, 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: 𝐷𝑥2𝛼𝑢(𝑥,𝑡)=𝐷𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝐷𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.11) where 𝛽=1/𝑞, 𝑝,𝑞,𝑟, 1<2𝛼2, 1<𝑝𝛽2, 0<𝑟𝛽1, 𝐷𝑥2𝛼𝐷𝛼𝑥𝐷𝛼𝑥, 𝐷𝑡𝑝𝛽𝐷𝛽𝑡𝐷𝛽𝑡𝐷𝛽𝑡 (𝑝 times), 𝐷𝑡𝑟𝛽𝐷𝛽𝑡𝐷𝛽𝑡𝐷𝛽𝑡 (𝑟 times), 𝐷𝛼𝑥, 𝐷𝛽𝑡 are Caputo fractional derivatives defined by (2.1), 𝑝+𝑟 is odd and 𝑢(0,𝑡)=𝐸𝛽𝑡𝛽,𝐷𝛼𝑥𝑢(𝑥,𝑡)𝑥=0=𝐸𝛽𝑡𝛽.(3.12) Applying generalized two-dimensional differential transform (2.4) with 𝑥0=0=𝑦0 to both sides of (3.11), (3.12) and using properties (c) and (d) we obtain 𝑈𝛼,𝛽(𝑘+2,)=Γ(𝛼𝑘+1)Γ(𝛼(𝑘+2)+1)Γ(𝛽(+𝑝)+1)𝑈Γ(𝛽+1)𝛼,𝛽+(𝑘,+𝑝)Γ(𝛽(+𝑟)+1)𝑈Γ(𝛽+1)𝛼,𝛽(𝑘,+𝑟)+𝑈𝛼,𝛽,(𝑘,)(3.13)𝑈𝛼,𝛽(0,)=(1)Γ(𝛽+1),𝑈𝛼,𝛽(1,)=(1)Γ(𝛼+1)Γ(𝛽+1),=0,1,2,.(3.14) Utilizing the recurrence relation (3.13), the transformed conditions (3.14) and the condition 𝑝+𝑟 is odd, we obtain 𝑈𝛼,𝛽(𝑘,)=(1)Γ(𝑘𝛼+1)Γ(𝛽+1),for𝑘,=0,1,2,.(3.15) Now, from (2.5), we have 𝑢(𝑥,𝑡)=𝑘=0=0𝑈𝛼,𝛽(𝑘,)𝑥𝛼𝑘𝑡𝛽.(3.16) Using the values of 𝑈𝛼,𝛽(𝑘,) from (3.15) in (3.16), the exact solution of homogeneous space-time fractional telegraph (3.11) is obtained as 𝑢(𝑥,𝑡)=𝐸𝛼(𝑥𝛼)𝐸𝛽𝑡𝛽.(3.17)

Remark 3.3. (1) Setting 𝑞=1, 𝑝=2, 𝑟=1, (3.11) reduces to space fractional telegraph equation 𝐷𝑥2𝛼𝑢(𝑥,𝑡)=𝐷2𝑡𝑢(𝑥,𝑡)+𝐷𝑡𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.18) with solution 𝑢(𝑥,𝑡)=𝐸𝛼(𝑥𝛼)𝑒𝑡.(3.19)
(2) Setting 𝛼=1, (3.11) reduces to time fractional telegraph equation: 𝐷2𝑥𝑢(𝑥,𝑡)=𝐷𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝐷𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.20) with solution 𝑢(𝑥,𝑡)=𝑒𝑥𝐸𝛽𝑡𝛽.(3.21)
(3) Setting 𝛼=1, 𝑞=1, 𝑝=2, 𝑟=1, (3.11) reduces to classical telegraph equation: 𝐷2𝑥𝑢(𝑥,𝑡)=𝐷2𝑡𝑢(𝑥,𝑡)+𝐷𝑡𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.22) with solution 𝑢(𝑥,𝑡)=𝑒𝑥𝑡,(3.23) 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: 𝐷𝑥2𝛼𝑢(𝑥,𝑡)=𝐷𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝐷𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡)2𝐸𝛼(𝑥𝛼)𝐸𝛽𝑡𝛽,0<𝑥<1,𝑡>0,(3.24) where 𝛽=1/𝑞, 𝑝,𝑞,𝑟, 1<2𝛼2, 1<𝑝𝛽2, 0<𝑟𝛽1, 𝐷𝑥2𝛼𝐷𝛼𝑥𝐷𝛼𝑥, 𝐷𝑡𝑝𝛽𝐷𝛽𝑡𝐷𝛽𝑡𝐷𝛽𝑡 (𝑝 times), 𝐷𝑡𝑟𝛽𝐷𝛽𝑡𝐷𝛽𝑡𝐷𝛽𝑡 (𝑟 times), 𝐷𝛼𝑥, 𝐷𝛽𝑡 are Caputo fractional derivatives defined by (2.1), 𝑝 and 𝑟 are even and 𝑢(0,𝑡)=𝐸𝛽𝑡𝛽,𝐷𝛼𝑥𝑢(𝑥,𝑡)𝑥=0=𝐸𝛽𝑡𝛽.(3.25) Applying generalized two-dimensional differential transform (2.4) with 𝑥0=0=𝑦0 to both sides of (3.24), (3.25), and using properties (c) and (d) we obtain 𝑈𝛼,𝛽(𝑘+2,)=Γ(𝛼𝑘+1)Γ(𝛼(𝑘+2)+1)Γ(𝛽(+𝑝)+1)𝑈Γ(𝛽+1)𝛼,𝛽(𝑘,+𝑝)+Γ(𝛽(+𝑟)+1)𝑈Γ(𝛽+1)𝛼,𝛽(𝑘,+𝑟)+𝑈𝛼,𝛽(𝑘,)2(1),Γ(𝛼𝑘+1)Γ(𝛽+1)(3.26)𝑈𝛼,𝛽(0,)=(1)Γ(𝛽+1),𝑈𝛼,𝛽(1,)=(1)Γ(𝛼+1)Γ(𝛽+1),=0,1,2,.(3.27) Utilizing the recurrence relation (3.26) and the transformed conditions (3.27), we obtain 𝑈𝛼,𝛽(𝑘,)=(1)Γ(𝑘𝛼+1)Γ(𝛽+1),𝑘,=0,1,2,.(3.28) Now from (2.5), we have 𝑢(𝑥,𝑡)=𝑘=0=0𝑈𝛼,𝛽(𝑘,)𝑥𝛼𝑘𝑡𝛽.(3.29) Using the values of 𝑈𝛼,𝛽(𝑘,) from (3.28) in (3.29), the exact solution of non-homogeneous space-time fractional telegraph (3.24) is obtained as 𝑢(𝑥,𝑡)=𝐸𝛼(𝑥𝛼)𝐸𝛽𝑡𝛽.(3.30)

Remark 3.5. (1) Setting 𝑞=2, 𝑝=4, 𝑟=2, (3.24) reduces to non-homogeneous space fractional telegraph equation: 𝐷𝑥2𝛼𝑢(𝑥,𝑡)=𝐷2𝑡𝑢(𝑥,𝑡)+𝐷𝑡𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡)2𝐸𝛼(𝑥𝛼)𝑒𝑡,0<𝑥<1,𝑡>0,(3.31) with solution 𝑢(𝑥,𝑡)=𝐸𝛼(𝑥𝛼)𝑒𝑡.(3.32)
(2) Setting 𝛼=1, (3.24) reduces to non-homogeneous time fractional telegraph equation: 𝐷2𝑥𝑢(𝑥,𝑡)=𝐷𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝐷𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡)2𝑒𝑥𝐸𝛽𝑡𝛽,0<𝑥<1,𝑡>0,(3.33) with solution 𝑢(𝑥,𝑡)=𝑒𝑥𝐸𝛽𝑡𝛽.(3.34)
(3) Setting 𝛼=1, 𝑞=2, 𝑝=4, 𝑟=2, (3.24) reduces to non-homogeneous telegraph equation: 𝐷2𝑥𝑢(𝑥,𝑡)=𝐷2𝑡𝑢(𝑥,𝑡)+𝐷𝑡𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡)2𝑒𝑥𝐸1/2𝑡1/2,0<𝑥<1,𝑡>0,(3.35) with solution 𝑢(𝑥,𝑡)=𝑒𝑥𝐸1/2𝑡1/2.(3.36)

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.

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