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Different initial and boundary value problems for the equation of vibrationsof rods (also called Fresnel equation) are solved by exploiting the connectionwith Brownian motion and the heat equation. The analysis of the fractionalversion (of order $\nu$) of the Fresnel equation is also performed and, indetail, some specific cases, like $\nu=1/2$, 1/3, 2/3, are analyzed. By meansof 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 propertiesexamined. The equation of vibrations of plates is considered and the case ofcircular vibrating disks $C_R$ is investigated by applying the methods ofplanar orthogonally reflecting Brownian motion within $C_R$. The composition ofF with reflecting Brownian motion $B$ yields the law of biquadratic heatequation 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