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 C(R) is investigated by applying the methods of planar orthogonally reflecting Brownian motion within C(R). 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 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.

Vibrations and Fractional Vibrations of Rods, Plates and Fresnel Pseudo-Processes / Orsingher, Enzo; D'Ovidio, Mirko. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - 145:1(2011), pp. 143-174. [10.1007/s10955-011-0309-5]

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

ORSINGHER, Enzo;D'OVIDIO, MIRKO
2011

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 C(R) is investigated by applying the methods of planar orthogonally reflecting Brownian motion within C(R). 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 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.
2011
brownian motion; elastic brownian motion; fractional diffusions; fresnel function; higher-order heat equations; inversion radius; mittag-leffler function; schrodinger equation; schrödinger equation; vibrations of plates; wright function
01 Pubblicazione su rivista::01a Articolo in rivista
Vibrations and Fractional Vibrations of Rods, Plates and Fresnel Pseudo-Processes / Orsingher, Enzo; D'Ovidio, Mirko. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - 145:1(2011), pp. 143-174. [10.1007/s10955-011-0309-5]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/386359
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