Generalized Differential Transform Method to Space-Time Fractional Telegraph Equation

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 [1–4]. 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 [5–7], homotopy perturbation method [8, 9], homotopy analysis method [10], variational iteration method [11, 12], matrix method [13], and differential transform method [14–16]. 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 [14–16] 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]𝑢_𝑡𝑡−𝑐²𝑢_𝑥𝑥+𝑎𝑢_𝑡+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]: 𝐷^𝛼𝑎𝑓(𝑥)=1Γ(𝑚−𝛼)^𝑥𝑎𝑓^(𝑚)(𝜉)(𝑥−𝜉)^{𝛼−𝑚+1}𝑑𝜉,(𝑚−1<𝛼≤𝑚),𝑚∈ℕ.(2.1)

Definition 2.2. The Mittag-Leffler function which is a generalization of exponential function is defined as [31]: 𝐸_𝛼(𝑧)=^∞_𝑛=0𝑧^𝑛Γ(𝛼𝑛+1)(𝛼∈𝐶,𝑅(𝛼)>0).(2.2) A further generalization of (2.2) is given in the form [32] 𝐸_𝛼,𝛽(𝑧)=^∞_𝑛=0𝑧^𝑛Γ(𝛼𝑛+𝛽);(𝛼,𝛽∈𝐶,𝑅(𝛼)>0,𝑅(𝛽)>0).(2.3) For 𝛼=1,𝐸_𝛼(𝑧) reduces to 𝑒^𝑧.

Definition 2.3. Generalized two-dimensional differential transform [14–16] 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 𝑈_𝛼,𝛽(𝑘,ℎ)=1Γ(𝛼𝑘+1)Γ(𝛽ℎ+1)𝐷^𝛼𝑥0^𝑘𝐷^𝛽𝑦0^ℎ𝑢(𝑥,𝑦)_{(𝑥0,𝑦0)},(2.4) where 0<𝛼, 𝛽≤1(𝐷^𝛼𝑥₀)^𝑘=𝐷^𝛼𝑥₀⋯𝐷^𝛼𝑥₀⋯𝐷^𝛼𝑥₀ (𝑘 times), 𝐷^𝛼𝑥₀ is defined by (2.1) and 𝑈_𝛼,𝛽(𝑘,ℎ) is the transformed function.

The generalized differential transform inverse of 𝑈_𝛼,𝛽(𝑘,ℎ) is given by𝑢(𝑥,𝑦)=^∞_∞𝑘=0_ℎ=0𝑈_𝛼,𝛽(𝑘,ℎ)𝑥−𝑥₀^𝑘𝛼𝑦−𝑦₀^ℎ𝛽.(2.5)

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

Let 𝑈_𝛼,𝛽(𝑘,ℎ), 𝑉_𝛼,𝛽(𝑘,ℎ) and 𝑊_𝛼,𝛽(𝑘,ℎ) 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 𝑈_𝛼,𝛽(𝑘,ℎ)=𝑉_𝛼,𝛽(𝑘,ℎ)±𝑊_𝛼,𝛽(𝑘,ℎ),(b)if 𝑢(𝑥,𝑦)=𝑎𝑣(𝑥,𝑦), 𝑎 is constant, then 𝑈_𝛼,𝛽(𝑘,ℎ)=𝑎𝑉_𝛼,𝛽(𝑘,ℎ),(c)if 𝑢(𝑥,𝑦)=𝐷^𝛾𝑥₀𝑣(𝑥,𝑦) where 𝑚−1<𝛾≤𝑚, 𝑚∈ℕ then 𝑈_𝛼,𝛽(𝑘,ℎ)=(Γ(𝛼𝑘+𝛾+1)/Γ(𝛼𝑘+1))𝑉_𝛼,𝛽(𝑘+𝛾/𝛼,ℎ),(d)if 𝑢(𝑥,𝑦)=𝐷^𝜇𝑦₀𝑣(𝑥,𝑦) where 𝑛−1<𝜇≤𝑛, 𝑛∈ℕ, then 𝑈_𝛼,𝛽(𝑘,ℎ)=(Γ(𝛽ℎ+𝜇+1)/Γ(𝛽ℎ+1))𝑉_𝛼,𝛽(𝑘,ℎ+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:𝑐²𝐷^𝑥2𝛼𝑢(𝑥,𝑡)=𝐷^𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝑎𝐷^𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑏𝑢(𝑥,𝑡)+𝑓(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.1) where 𝛽=1/𝑞, 𝑝,𝑞,𝑟∈ℕ, 1<2𝛼≤2, 1<𝑝𝛽≤2, 0<𝑟𝛽≤1, 𝐷^𝑥2𝛼≡𝐷^𝛼𝑥𝐷^𝛼𝑥, 𝐷^𝑡𝑝𝛽≡𝐷^𝛽𝑡𝐷^𝛽𝑡⋯𝐷^𝛽𝑡 (𝑝 times), 𝐷^𝑡𝑟𝛽≡𝐷^𝛽𝑡𝐷^𝛽𝑡⋯𝐷^𝛽𝑡 (𝑟 times), 𝐷^𝛼𝑥, 𝐷^𝛽𝑡 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: 𝐷^𝑥3/2𝑢(𝑥,𝑡)=𝐷^𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝐷^𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.2) where 𝛽=1/𝑞, 𝑝,𝑞,𝑟∈ℕ, 1<𝑝𝛽≤2, 0<𝑟𝛽≤1, 𝐷^𝑥3/2≡(𝐷^𝑥1/2)³, 𝐷^𝑡𝑝𝛽≡𝐷^𝛽𝑡𝐷^𝛽𝑡⋯𝐷^𝛽𝑡 (𝑝 times), 𝐷^𝑡𝑟𝛽≡𝐷^𝛽𝑡𝐷^𝛽𝑡⋯𝐷^𝛽𝑡 (𝑟 times), 𝐷^𝛼𝑥, 𝐷^𝛽𝑡 are Caputo fractional derivatives defined by (2.1), 𝑝+𝑟 is odd and 𝑢(0,𝑡)=𝐸_𝛽−𝑡^𝛽,𝑢_𝑥(0,𝑡)=𝐸_𝛽−𝑡^𝛽.(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 𝑈_1/2,𝛽(𝑘+3,ℎ)=Γ((𝑘/2)+1)Γ(((𝑘+3)/2)+1)Γ(𝛽Γ((ℎ𝛽ℎ++𝑝)1+)1)𝑈^1/2,𝛽+Γ(𝛽(ℎ+𝑟)+1)(𝑘,ℎ+𝑝)Γ(𝛽ℎ+1)𝑈^1/2,𝛽(𝑘,ℎ+𝑟)+𝑈_1/2,𝛽(𝑘,ℎ),(3.4)𝑈_1/2,𝛽(0,ℎ)=(−1)^ℎΓ(𝛽ℎ+1),𝑈^1/2,𝛽𝑈(1,ℎ)=0,_1/2,𝛽(2,ℎ)=(−1)^ℎΓ(𝛽ℎ+1),ℎ=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,ℎ=0,1,2,…𝑈_1/2,𝛽(3𝑙,ℎ)=(−1)^ℎ𝑈Γ((3/2)𝑙+1)Γ(𝛽ℎ+1),(3.6)_1/2,𝛽𝑈(3𝑙+1,ℎ)=0,(3.7)_1/2,𝛽(3𝑙+2,ℎ)=(−1)^ℎΓ((3/2)𝑙+2)Γ(𝛽ℎ+1).(3.8) Now, from (2.5), we have 𝑢(𝑥,𝑡)=^∞_∞𝑘=0_ℎ=0𝑈_1/2,𝛽(𝑘,ℎ)𝑥^𝑘/2𝑡^𝛽ℎ.(3.9) Using the values of 𝑈_1/2,𝛽(𝑘,ℎ) from (3.5)–(3.8) in (3.9), the exact solution of space-time fractional telegraph (3.2) is obtained as 𝑢(𝑥,𝑡)=𝐸_3/2𝑥^3/2+𝑥𝐸_3/2,2𝑥^3/2𝐸_𝛽−𝑡^𝛽,(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: 𝐷^𝑥2𝛼𝑢(𝑥,𝑡)=𝐷^𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝐷^𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.11) where 𝛽=1/𝑞, 𝑝,𝑞,𝑟∈ℕ, 1<2𝛼≤2, 1<𝑝𝛽≤2, 0<𝑟𝛽≤1, 𝐷^𝑥2𝛼≡𝐷^𝛼𝑥𝐷^𝛼𝑥, 𝐷^𝑡𝑝𝛽≡𝐷^𝛽𝑡𝐷^𝛽𝑡⋯𝐷^𝛽𝑡 (𝑝 times), 𝐷^𝑡𝑟𝛽≡𝐷^𝛽𝑡𝐷^𝛽𝑡⋯𝐷^𝛽𝑡 (𝑟 times), 𝐷^𝛼𝑥, 𝐷^𝛽𝑡 are Caputo fractional derivatives defined by (2.1), 𝑝+𝑟 is odd and 𝑢(0,𝑡)=𝐸_𝛽−𝑡^𝛽,𝐷^𝛼𝑥𝑢(𝑥,𝑡)∣_𝑥=0=𝐸_𝛽−𝑡^𝛽.(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 𝑈_𝛼,𝛽(𝑘+2,ℎ)=Γ(𝛼𝑘+1)Γ(𝛼(𝑘+2)+1)Γ(𝛽Γ((ℎ𝛽ℎ++𝑝)1+)1)𝑈^𝛼,𝛽+Γ(𝛽(ℎ+𝑟)+1)(𝑘,ℎ+𝑝)Γ(𝛽ℎ+1)𝑈^𝛼,𝛽(𝑘,ℎ+𝑟)+𝑈_𝛼,𝛽(𝑘,ℎ),(3.13)𝑈_𝛼,𝛽(0,ℎ)=(−1)^ℎΓ(𝛽ℎ+1),𝑈^𝛼,𝛽(1,ℎ)=(−1)^ℎΓ(𝛼+1)Γ(𝛽ℎ+1),ℎ=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 𝑈_𝛼,𝛽(𝑘,ℎ)=(−1)^ℎΓ(𝑘𝛼+1)Γ(ℎ𝛽+1),for𝑘,ℎ=0,1,2,….(3.15) Now, from (2.5), we have 𝑢(𝑥,𝑡)=^∞_∞𝑘=0_ℎ=0𝑈_𝛼,𝛽(𝑘,ℎ)𝑥^𝛼𝑘𝑡^𝛽ℎ.(3.16) Using the values of 𝑈_𝛼,𝛽(𝑘,ℎ) from (3.15) in (3.16), the exact solution of homogeneous space-time fractional telegraph (3.11) is obtained as 𝑢(𝑥,𝑡)=𝐸_𝛼(𝑥^𝛼)𝐸_𝛽−𝑡^𝛽.(3.17)

Remark 3.3. (1) Setting 𝑞=1, 𝑝=2, 𝑟=1, (3.11) reduces to space fractional telegraph equation 𝐷^𝑥2𝛼𝑢(𝑥,𝑡)=𝐷^2𝑡𝑢(𝑥,𝑡)+𝐷_𝑡𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.18) with solution 𝑢(𝑥,𝑡)=𝐸_𝛼(𝑥^𝛼)𝑒^−𝑡.(3.19)
(2) Setting 𝛼=1, (3.11) reduces to time fractional telegraph equation: 𝐷^2𝑥𝑢(𝑥,𝑡)=𝐷^𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝐷^𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.20) with solution 𝑢(𝑥,𝑡)=𝑒^𝑥𝐸_𝛽−𝑡^𝛽.(3.21)
(3) Setting 𝛼=1, 𝑞=1, 𝑝=2, 𝑟=1, (3.11) reduces to classical telegraph equation: 𝐷^2𝑥𝑢(𝑥,𝑡)=𝐷^2𝑡𝑢(𝑥,𝑡)+𝐷_𝑡𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡),0<𝑥<1,𝑡>0,(3.22) with solution 𝑢(𝑥,𝑡)=𝑒^𝑥−𝑡,(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: 𝐷^𝑥2𝛼𝑢(𝑥,𝑡)=𝐷^𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝐷^𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡)−2𝐸_𝛼(𝑥^𝛼)𝐸_𝛽−𝑡^𝛽,0<𝑥<1,𝑡>0,(3.24) where 𝛽=1/𝑞, 𝑝,𝑞,𝑟∈ℕ, 1<2𝛼≤2, 1<𝑝𝛽≤2, 0<𝑟𝛽≤1, 𝐷^𝑥2𝛼≡𝐷^𝛼𝑥𝐷^𝛼𝑥, 𝐷^𝑡𝑝𝛽≡𝐷^𝛽𝑡𝐷^𝛽𝑡⋯𝐷^𝛽𝑡 (𝑝 times), 𝐷^𝑡𝑟𝛽≡𝐷^𝛽𝑡𝐷^𝛽𝑡⋯𝐷^𝛽𝑡 (𝑟 times), 𝐷^𝛼𝑥, 𝐷^𝛽𝑡 are Caputo fractional derivatives defined by (2.1), 𝑝 and 𝑟 are even and 𝑢(0,𝑡)=𝐸_𝛽−𝑡^𝛽,𝐷^𝛼𝑥𝑢(𝑥,𝑡)∣_𝑥=0=𝐸_𝛽−𝑡^𝛽.(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 𝑈_𝛼,𝛽(𝑘+2,ℎ)=Γ(𝛼𝑘+1)Γ(𝛼(𝑘+2)+1)Γ(𝛽Γ((ℎ𝛽ℎ++𝑝)1+)1)𝑈^𝛼,𝛽(𝑘,ℎ+𝑝)+Γ(𝛽(ℎ+𝑟)+1)Γ(𝛽ℎ+1)𝑈^𝛼,𝛽+𝑈_𝛼,𝛽(𝑘,ℎ+𝑟)(𝑘,ℎ)−2(−1)^ℎΓ(𝛼𝑘+1)Γ(𝛽ℎ+1),(3.26)𝑈_𝛼,𝛽(0,ℎ)=(−1)^ℎΓ(𝛽ℎ+1),𝑈^𝛼,𝛽(1,ℎ)=(−1)^ℎΓ(𝛼+1)Γ(𝛽ℎ+1),ℎ=0,1,2,….(3.27) Utilizing the recurrence relation (3.26) and the transformed conditions (3.27), we obtain 𝑈_𝛼,𝛽(𝑘,ℎ)=(−1)^ℎΓ(𝑘𝛼+1)Γ(ℎ𝛽+1),𝑘,ℎ=0,1,2,….(3.28) Now from (2.5), we have 𝑢(𝑥,𝑡)=^∞_∞𝑘=0_ℎ=0𝑈_𝛼,𝛽(𝑘,ℎ)𝑥^𝛼𝑘𝑡^𝛽ℎ.(3.29) Using the values of 𝑈_𝛼,𝛽(𝑘,ℎ) from (3.28) in (3.29), the exact solution of non-homogeneous space-time fractional telegraph (3.24) is obtained as 𝑢(𝑥,𝑡)=𝐸_𝛼(𝑥^𝛼)𝐸_𝛽−𝑡^𝛽.(3.30)

Remark 3.5. (1) Setting 𝑞=2, 𝑝=4, 𝑟=2, (3.24) reduces to non-homogeneous space fractional telegraph equation: 𝐷^𝑥2𝛼𝑢(𝑥,𝑡)=𝐷^2𝑡𝑢(𝑥,𝑡)+𝐷_𝑡𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡)−2𝐸_𝛼(𝑥^𝛼)𝑒^−𝑡,0<𝑥<1,𝑡>0,(3.31) with solution 𝑢(𝑥,𝑡)=𝐸_𝛼(𝑥^𝛼)𝑒^−𝑡.(3.32)
(2) Setting 𝛼=1, (3.24) reduces to non-homogeneous time fractional telegraph equation: 𝐷^2𝑥𝑢(𝑥,𝑡)=𝐷^𝑡𝑝𝛽𝑢(𝑥,𝑡)+𝐷^𝑡𝑟𝛽𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡)−2𝑒^𝑥𝐸_𝛽−𝑡^𝛽,0<𝑥<1,𝑡>0,(3.33) with solution 𝑢(𝑥,𝑡)=𝑒^𝑥𝐸_𝛽−𝑡^𝛽.(3.34)
(3) Setting 𝛼=1, 𝑞=2, 𝑝=4, 𝑟=2, (3.24) reduces to non-homogeneous telegraph equation: 𝐷^2𝑥𝑢(𝑥,𝑡)=𝐷^2𝑡𝑢(𝑥,𝑡)+𝐷_𝑡𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡)−2𝑒^𝑥𝐸_1/2−𝑡^1/2,0<𝑥<1,𝑡>0,(3.35) with solution 𝑢(𝑥,𝑡)=𝑒^𝑥𝐸_1/2−𝑡^1/2.(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.