Poincarè半平面上广义分数阶扩散方程的一个注记
A note on generalized fractional diffusion equations on Poincarè half plane
摘要：
本文研究Poincarè半平面$\mathbb{H}_2^+$上的广义时间分数阶扩散方程.这里考虑的时间分数阶算子是一个函数关于另一个函数的分数阶导数，它可以从经典的Caputo导数出发，基本上通过确定性的变量变化来获得。我们得到了$\mathbb{H}_2^+$上广义扩散方程的基本解的显式表示，并提供了与时变双曲布朗运动相关的概率解释。最后，我们包括一个明确的结果关于非线性的情况下，承认分离变量的解决方案。
英文摘要：
In this paper we study generalized time-fractional diffusion equations on the Poincarè 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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