Vibrations and Fractional Vibrations of Rods, Plates and Fresnel Pseudo-Processes

TLDR: In this paper, the initial and boundary value problems for the equation of vibrations of rods (also called Fresnel equation) are solved by exploiting the connection with Brownian motion and the heat equation.

Abstract: Different initial and boundary value problems for the equation of vibrations of rods (also called Fresnel equation) are solved by exploiting the connection with Brownian motion and the heat equation. The equation of vibrations of plates is considered and the case of circular vibrating disks CR is investigated by applying the methods of planar orthogonally reflecting Brownian motion within CR. The analysis of the fractional version (of order ν) of the Fresnel equation is also performed and, in detail, some specific cases, like ν=1/2, 1/3, 2/3, are analyzed. By means of the fundamental solution of the Fresnel equation, a pseudo-process F(t), t>0 with real sign-varying density is constructed and some of its properties examined. The composition of F with reflecting Brownian motion B yields the law of biquadratic heat equation while the composition of F with the first passage time Tt of B produces a genuine probability law strictly connected with the Cauchy process.