Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes

TL;DR

This paper explores vibrations of rods and plates through classical and fractional equations, connecting them with Brownian motion and pseudo-processes, and analyzes specific cases and their probabilistic properties.

Contribution

It introduces a novel approach linking the Fresnel vibration equations with Brownian motion and pseudo-processes, including fractional versions and their probabilistic interpretations.

Findings

Constructed a pseudo-process with sign-varying density from the Fresnel equation.

Analyzed fractional Fresnel equations for specific orders like 1/2, 1/3, 2/3.

Connected vibration equations with Brownian motion and Cauchy processes.

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 analysis of the fractional version (of order $\nu$) of the Fresnel equation is also performed and, in detail, some specific cases, like $\nu=1/2$, 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 equation of vibrations of plates is considered and the case of circular vibrating disks $C_R$ is investigated by applying the methods of planar orthogonally reflecting Brownian motion within $C_R$. 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 $T_t$ of $B$ produces a genuine probability law strictly connected with the Cauchy process.