Fundamental solutions of the fractional Fresnel equation in the real half-line

Abstract

The purpose of this paper is to offer a discussion on the fundamental solutions for the vibration fractional equation of rods semi-infinite (also called Fresnel equation). For this end, using the fractional derivative of Riemann–Liouville of order γ∈(1,2] with respect to the spatial variable, we solve boundary-value problems associated to the mentioned equation showing the connection with Brownian behavior and the heat equation. We obtain solutions for the fractional Fresnel equation using the unified transform method. In the intermediate of the investigation, we have solved generalized Cauchy problems associated with linear fractional Schrödinger equations with order γ.