Modeling mass transport using time-variable space-fractional Riesz derivative

In this work, we consider the model of a mass transport problem through polymeric membranes which is important in many applications especially drug delivery. The aim of this paper is to present a more accurate mathematical model to describe this phenomenon. We propose adopting fractional diffusion modeling for the concentration by using spatial fractional-order Riesz derivative. The numerical simulations for studying this phenomenon were carried out using an approach of finite element method and finite difference method. The error analysis and stability condition for this technique are derived. For the order of the fractional derivative, we studied three cases: the constant-order case, the piecewise continuous case, and the time-varying-order case. The results obtained show that using time-varying Riesz fractional derivative yields a more accurate description of the considered problem than both the classical diffusion model reported in the literature and the other fractional derivatives considered.