杆、板和菲涅耳伪过程的振动和分数振动
Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes
摘要：
利用布朗运动与热方程的联系，求解了杆振动方程（也称Fresnel方程）的不同初边值问题和边值问题。对菲涅耳方程的分数阶形式（$ν$阶）进行了分析，并详细分析了$ν=1/2$、1/3、2/3等具体情况。利用Fresnel方程的基本解，构造了一个具有实变号密度的伪过程$F(t)$，$t>0$，并研究了它的一些性质.本文考虑板的振动方程，并应用平面正交反射布朗运动的方法研究了圆振动圆盘$C_R$在$C_R$中的情况。F与反射布朗运动$B$的合成产生了双二次热方程定律，而$F$与$B$的首次通过时间$T_t$的合成产生了与柯西过程严格相关的真正概率定律。
英文摘要：
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 $ν$) 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 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.
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