  In this paper we study $g$-fractional diffusion on bounded domains in
$\mathbb{R}^d$ with absorbing boundary conditions. We show the explicit
representation of the solution and then we study the first passage time
distribution, showing the dependence on the particular choice of the function
$g$. Then, we specialize the analysis to the interesting case of a rectangular
domain. Finally we briefly discuss the connection of this general theory with
the physical application to the so-called fractional Dodson diffusion model
recently discussed in the literature.
