On fractional Cattaneo equation with partially reflecting boundaries

In this paper we study the time-fractional Cattaneo equation in a bounded domain with semi-reflecting conditions. In particular, we are able to find the Laplace transform of the probability density function of the absorption time and therefore the mean-time to absorption. We show the crucial role of the time-fractional formulation. Indeed, in this case we have that the mean-time to absorption diverges due to the fact that the generalized Cattaneo equation is based on the application of integral operators with a long-tail memory kernel. We also consider the time-fractional diffusion and wave limits behaviour, recovering some previous results obtained in the literature. Finally, a section is devoted to the generalized Cattaneo equation in unbounded domain. In this case we are able to discuss the characterization of the mean square displacement for short times and asymptotically by using the Fourier-Laplace transform of the solution.