Fundamental solution of the multi-dimensional time fractional telegraph equation

Abstract

In this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation with time-fractional derivatives of orders $\alpha \in ]0,1]$ and $\beta \in ]1,2]$ in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS expressed in terms of a multivariate Mittag-Leffler function in the Fourier domain. The Fourier inversion leads to a double Mellin-Barnes type integral representation and consequently to a H-function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is also obtained. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension $n$ and of the fractional parameters $\alpha$ and $\beta$.