Fractional Diffusion Problems with Reflecting Boundaries

Anomalous diffusive transport, described by fractional differential equations, arises in a large variety of physical problems. We consider a fractional diffusion equation subjected to reflecting boundary conditions. The formulation of these boundaries has sparked a controversial discussion, with questions arising about the most appropriate boundary from the physical point of view. Therefore, we start to present different physical formulations regarding the boundaries. Numerical methods are then proposed to solve these diffusive models, and it is shown how the presence of boundaries changes the general structure of the problem and of the numerical method, due to the non-locality of the problem. In the end, the impact of the different boundaries on the solutions is analysed.