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Title: Time-fractional telegraph equation with ψ-Hilfer derivatives
Author: Vieira, N.
Ferreira, M.
Rodrigues, M. M.
Keywords: Time-fractional telegraph equation
ψ-Hilfer fractional derivative
ψ-Laplace transform
Series and integral representations
Fractional moments
Probability density function
Issue Date: Sep-2022
Publisher: Elsevier
Abstract: This paper deals with the investigation of the solution of the time-fractional telegraph equation in higher dimensions with $\psi$-Hilfer fractional derivatives. By application of the Fourier and $\psi$-Laplace transforms the solution is derived in closed form in terms of bivariate Mittag-Leffler functions in the Fourier domain and in terms of convolution integrals involving Fox H-functions of two-variables in the space-time domain. A double series representation 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 fractional derivatives and the use of an arbitrary positive monotone increasing function $\psi$ in the kernel allows to encompass 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.
Peer review: yes
DOI: 10.1016/j.chaos.2022.112276
ISSN: 0960-0779
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Appears in Collections:CIDMA - Artigos
CHAG - Artigos

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