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

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 [

The fractional telegraph equation has recently been considered by many authors. Cascaval et al. [

In this paper, we consider the following time-fractional telegraph equation (TFTE)

For the TFTE (

TFTE in a whole-space domain (Cauchy problem)

TFTE in a half-space domain (Signaling problem)

TFTE in a bounded-space domain

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

We first focus our attention on (

Applying temporal Laplace and spatial Fourier transforms to (

To express the Green function, we recall two Laplace transform pairs and one Fourier transform pair,

Then the Fourier-Laplace transform of the Green function (

Going back to the space-time domain we obtain the relation

By the same technique, we can obtain the expression of

Going back to the space-time domain we obtain the relation

We can ensure that the green functions are nonnegative by the nonnegative prosperities of

In this section, we considered Problem

By the application of the Laplace transform to (

In this section we seek the solution of Problem

Taking the finite Sine transform of (

Applying the Laplace transform to (

We set

To inverse the Laplace transform for (

Then we obtain the pairs

So we inverse Laplace and finite Sine transform for (

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.

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).