Analytical Solution for the Time-Fractional Telegraph Equation

Abstract

We discuss and derive the analytical solution for three basic problems of the so-called time-fractional telegraph equation. The Cauchy and Signaling problems are solved by means of juxtaposition of transforms of the Laplace and Fourier transforms in variable t and x, respectively. the appropriate structures and negative prosperities for their Green functions are provided. The boundary problem in a bounded space domain is also solved by the spatial Sine transform and temporal Laplace transform, whose solution is given in the form of a series.

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Recommended by Jacek Rokicki

Department of Mathematics, School of Sciences, South China University of Technology, Guangzhou 510641, China

Received 24 April 2009; Accepted 14 October 2009

1. Introduction

Fractional differential equations (FDEs) have attracted in the recent years a considerable interest due to their frequent appearance in various fields and their more accurate models of systems under consideration provided by fractional derivatives. For example, fractional derivatives have been used successfully to model frequency dependent damping behavior of many viscoelastic materials. They are also used in modeling of many chemical processed, mathematical biology and many other problems in engineering. The history and a comprehensive treatment of FDEs are provided by Podlubny [1] and a review of some applications of FDEs are given by Mainardi [2].

The fractional telegraph equation has recently been considered by many authors. Cascaval et al. [3] discussed the time-fractional telegraph equations, dealing with well-posedness and presenting a study involving asymptotic by using the Riemann-Liouville approach. Orsingher and Beghin [4] discussed the time-fractional telegraph equation and telegraph processes with Brownian time, showing that some processes are governed by time-fractional telegraph equations. Chen et al. [5] also discussed and derived the solution of the time-fractional telegraph equation with three kinds of nonhomogeneous boundary conditions, by the method of separating variables. Orsingher and Zhao [6] considered the space-fractional telegraph equations, obtaining the Fourier transform of its fundamental solution and presenting a symmetric process with discontinuous trajectories, whose transition function satisfies the space-fractional telegraph equation. Momani [7] discussed analytic and approximate solutions of the space- and time-fractional telegraph differential equations by means of the so-called Adomian decomposition method. Camargo et al. [8] discussed the so-called general space-time fractional telegraph equations by the methods of differential and integral calculus, discussing the solution by means of the Laplace and Fourier transforms in variables t and x , respectively.

In this paper, we consider the following time-fractional telegraph equation (TFTE) [figure omitted; refer to PDF] where a,d are positive constants, 1/2<α≤1 , ^Dtβ is the fractional derivative defined in the Caputo sense: [figure omitted; refer to PDF] where f(t) is a continuous function. Properties and more details about the Caputo's fractional derivative also can be found in [1, 2].

For the TFTE (1.1), we will consider three basic problems with the following three kinds of initial and boundary conditions, respectively.

Problem 1.

TFTE in a whole-space domain (Cauchy problem) [figure omitted; refer to PDF]

Problem 2.

TFTE in a half-space domain (Signaling problem) [figure omitted; refer to PDF] [figure omitted; refer to PDF] and we set f(x,t)=0 in (1.1).

Problem 3.

TFTE in a bounded-space domain [figure omitted; refer to PDF] [figure omitted; refer to PDF] here we also set f(x,t)=0 in (1.1).

In this paper, we derive the analytical solutions of the previous three problems for the TFTE. The structure of the paper is as follows. In Section 2, by using the method of Laplace and Fourier transforms, the fundamental solution of Problem 1 is derived. In Section 3, by investigating the explicit relationships of the Laplace Transforms to the Green functions between Problems 1 and 2, the fundamental solution of the Problem 2 is also derived. The analytical solution of Problem 3 is presented in Section 4. Some conclusions are drawn in Section 5.

2. The Cauchy Problem for the TFTE

We first focus our attention on (1.1) in a whole-space domain, that is to say, Problem 1 will to be considered, which we refer to as the so-called Cauchy problem.

Applying temporal Laplace and spatial Fourier transforms to (1.1) and using the initial boundary conditions (1.3), we obtain the following nonhomogeneous differential equation: [figure omitted; refer to PDF] Then we derive [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [figure omitted; refer to PDF] By the Fourier transform pair [figure omitted; refer to PDF] we also have [figure omitted; refer to PDF] [figure omitted; refer to PDF] We invert the Fourier transform in (2.2) to obtain [figure omitted; refer to PDF] where _G1 (x,t), _G2 (x,t) is the corresponding Green function or fundamental solution obtained when [varphi](x)=δ(x), f(x)=0 and [varphi](x)=0, f(x,t)=δ(x)δ(t) respectively, which is characterized by (2.4) or (2.3).

To express the Green function, we recall two Laplace transform pairs and one Fourier transform pair, [figure omitted; refer to PDF] where _Mβ denotes the so-called M function (of the Wright type) of order β , which is defined [figure omitted; refer to PDF] Mainardi, see, for example, [9] has showed that _Mβ (z) is positive for z>0 , the other general properties can be found in some references (see [1, 9-11] e.g,).

_wβ (0<β<1) denotes the one-sided stable (or Lévy) probability density which can be explicitly expressed by Fox function [12] [figure omitted; refer to PDF]

Then the Fourier-Laplace transform of the Green function (2.4) can be rewritten in integral form [figure omitted; refer to PDF]

Going back to the space-time domain we obtain the relation [figure omitted; refer to PDF]

By the same technique, we can obtain the expression of _G2 (x,t) : [figure omitted; refer to PDF]

Going back to the space-time domain we obtain the relation [figure omitted; refer to PDF]

We can ensure that the green functions are nonnegative by the nonnegative prosperities of ^F1(β) ,^F2(β) ,_F3 .

3. The Solution for the TFTE in Half-Space Domain (Signaling Problems)

In this section, we considered Problem 2, defined in a half-space domain, which we refer to as the so-called Signaling problem.

By the application of the Laplace transform to (1.1) and (1.5) with f≡0 and the initial condition (1.4), we get [figure omitted; refer to PDF] with the solution [figure omitted; refer to PDF] where _Gs (x,t) is the Green function or fundamental solution of the Signaling problem obtained when g(x)=δ(x) , which is characterized by [figure omitted; refer to PDF] The inverse Laplace transform of (3.2) gives the solution of Problem 2 [figure omitted; refer to PDF] From (2.6), (2.7) and (3.3), we recognize the relation [figure omitted; refer to PDF] Returning to the space-time domain we obtain the relation [figure omitted; refer to PDF] Then we can obtain a representation for _Gs (x,t) and prove the negative prosperities.

4. The Solution of the TFTE in a Bounded-Space Domain

In this section we seek the solution of Problem 3, which is defined in a bounded domain.

Taking the finite Sine transform of (1.1) with f=0 , and applying the boundary conditions (1.7), we obtain [figure omitted; refer to PDF] where n is a wave number, and [figure omitted; refer to PDF] is the finite Sine transform of u(x,t) .

Applying the Laplace transform to (4.1) and using the initial conditions (1.6), we obtain [figure omitted; refer to PDF]

We set _λ± =-a±^a2 -^(ndπ/L)2 , then [figure omitted; refer to PDF]

To inverse the Laplace transform for (4.3), we recall the known Laplace transform pair [figure omitted; refer to PDF] where _Eα,β (z) is the so-called two-parameter Mittag-Leffler function, which is defined as follows: [figure omitted; refer to PDF] and we note _Eα,1 =_Eα .

Then we obtain the pairs [figure omitted; refer to PDF] where _c1 =_λ+ /(_λ+ -_λ- ), _c2 =_λ- /(_λ+ -_λ- ) .

So we inverse Laplace and finite Sine transform for (4.3) to obtain [figure omitted; refer to PDF]

5. Conclusions

In this paper we have considered the time-fractional telegraph equation. The fundamental solution for the Cauchy problem in a whole-space domain and Signaling problem in a half-space domain is obtained by using Fourier-Laplace transforms and their inverse transforms. The appropriate structures and negative prosperities for the Green functions are provided. On the other hand, the solution in the form of a series for the boundary problem in a bounded-space domain is derived by the Sine-Laplace transforms method.

Acknowledgments

This work is supported by NSF of China (Tianyuan Fund for Mathematics, no. 10726061), by NSF of Guangdong Province (no. 07300823), and by the Research Fund for the Doctoral Program of Higher Education of China (for new teachers, no. 20070561040).

References

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