Time-fractional telegraph equations and telegraph processes with brownian time

Abstract

We study the fundamental solutions to time-fractional telegraph equations of order 2α. We are able to obtain the Fourier transform of the solutions for any α and to give a representation of their inverse, in terms of stable densities. For the special case α = 1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.

Fractional telegrapher's equation from fractional persistent random walks

“…and recalling that L −1 {s −δ } = t δ−1 / (δ) (δ > 0) [57], we get the exact expression for the mean square displacement [48] …”

“…During the last decade a number of works have appeared in the mathematics literature, particularly those of Orshinger and collaborators [47][48][49], analyzing mathematical and other formal properties of the FTE. However, the fractional equation is set in an ad hoc fashion by replacing the ordinary derivatives that appear in the TE by fractional derivatives which, in addition, may be of various types [49].…”

Three-dimensional telegrapher's equation and its fractional generalization

“…However, for the sake of simplicity and briefness, we have not treated them here, and we will deal with these questions in a future work. In any case, we address the interested reader to the works of Mainardi and collaborators [46,47,62,64] on solutions for fractional diffusion and fractional wave-diffusion equations and to Orsingher and collaborators [65][66][67] on several kinds of solutions to the FTE.…”

“…[50,66] for further details). From this exact expression it easily is shown that the mean square displacement exists and that is approximated by…”

Fractional Difference Approximations for Time-Fractional Telegraph Equation

“…What we will discuss is the following time-fractional telegraph equation [13]: As for the boundary condition, we consider the Dirichlet boundary condition, …”