Fractional diffusion and reflective boundary condition

Anomalous diffusive transport arises in a large diversity of disordered media. Stochastic formulations in terms of continuous time random walks (CTRW) with transition probability densities presenting spatial and/or time diverging moments were developed to account for anomalous behaviours. Many CTRWs in infinite media were shown to correspond, on the macroscopic scale, to diffusion equations sometimes involving derivatives of non-integer order. A wide class of CTRWs with symmetric Lévy distribution of jumps and finite mean waiting time leads, in the macroscopic limit, to space-time fractional equations that account for super diffusion and involve an operator, which is non-local in space. Due to non-locality, the boundary condition results in modifying the large-scale model. We are studying here the diffusive limit of CTRWs, generalizing Lévy flights in a semi-infinite medium, limited by a reflective barrier. We obtain space-time fractional diffusion equations that differ from the infinite medium in the kernel of the fractional derivative w.r.t. space.