Time-fractional telegraph equation with 111-Hilfer derivatives

This paper deals with the investigation of the solution of the time-fractional telegraph equation in higher dimen-sions with 111-Hilfer fractional derivatives. By application of the Fourier and 111-Laplace transforms the solution is derived in closed form in terms of bivariate Mittag-Leffler functions in the Fourier domain and in terms of convo-lution integrals involving Fox H-functions of two-variables in the space-time domain. A double series represen-tation of the first fundamental solution is deduced for the case of odd dimension. The results derived here are of general nature since our fractional derivatives allow to interpolate between Riemann-Liouville and Caputo frac-tional derivatives and the use of an arbitrary positive monotone increasing function 111 in the kernel allows to en-compass most of the fractional derivatives in the literature. In the one dimensional case, we prove the conditions under which the first fundamental solution of our equation can be interpreted as a spatial probability density function evolving in time, generalizing the results of Orsingher and Beghin (2004). Some plots of the fundamental solutions for different fractional derivatives are presented and analysed, and particular cases are addressed to show the consistency of our results. 2010 MSC: 35R11, 26A33, 35A08, 35A22, 35C15, 60G22.(c) 2022 Elsevier Ltd. All rights reserved.