Abstract
In this paper we study linear fractional differential equations involving tempered Caputo-type derivatives in the hyperbolic space. We consider in detail the three-dimensional case for its simple and useful structure. We also discuss the probabilistic meaning of our results in relation to the distribution of an hyperbolic Brownian motion time-changed with the inverse of a tempered stable subordinator. The generalization to an arbitrary dimension n can be easily obtained. We also show that it is possible to construct a particular solution for the non-linear porous-medium type tempered equation by using elementary functions.