关于庞卡罗半平面上广义分数扩散方程的一个注记

R. Garra, F. Maltese, E. Orsingher
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引用次数: 2

摘要

本文研究了庞加莱半平面$\mathbb{H}_2^+$上的广义时间分数扩散方程。这里考虑的时间分数算子是一个函数对另一个函数的分数阶导数,它可以从经典的卡普托导数开始,基本上是通过变量的确定性变化来获得。得到了$\mathbb{H}_2^+$上广义扩散方程基本解的显式表示,并给出了时变双曲布朗运动的概率解释。最后,我们给出了一个关于非线性情况下允许分离变量解的显式结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A note on generalized fractional diffusion equations on Poincaré half plane
In this paper we study generalized time-fractional diffusion equations on the Poincar\`e half plane $\mathbb{H}_2^+$. The time-fractional operators here considered are fractional derivatives of a function with respect to another function, that can be obtained by starting from the classical Caputo-derivatives essentially by means of a deterministic change of variable. We obtain an explicit representation of the fundamental solution of the generalized-diffusion equation on $\mathbb{H}_2^+$ and provide a probabilistic interpretation related to the time-changed hyperbolic Brownian motion. We finally include an explicit result regarding the non-linear case admitting a separating variable solution.
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