5 - Fractional Diffusion Equations

Summary

In this chapter, we consider the influence of the surfaces or membranes on diffusive processes. The main aim is to investigate how the surface may modify the diffusive process of a system governed by a fractional diffusion equation.

In the first part of the chapter, we analyse the one-dimensional problem characterised by time-dependent boundary conditions, showing how they influence the diffusive process in the system for an arbitrary initial condition, i.e., the quantities related to the diffusion process, such as the first passage time, which may have an anomalous behaviour. A similar analysis is carried out for the two-dimensional case with inhomogeneous and time-dependent boundary conditions. These results show the potential of this formalism to analyse other physical scenarios, such as describing the molecular orientation and the anchoring problem in liquid crystals confined to a cylindrical region, taking into account the adsorption phenomena at the interfaces.

The second part of the chapter is dedicated to investigating situations in which the processes occurring on the surface are coupled to the bulk dynamics by means of the boundary conditions. As a first application, we consider a surface in which, besides the adsorption–desorption process, a reaction process may occur and the system presents anomalous diffusion behaviour. Another application refers to the transport through a membrane of definite thickness, for which the processes occurring on the surface also couple with the diffusion equations governing the bulk dynamics. In all cases, the system may exhibit an anomalous diffusive behaviour for which surface effects play a remarkable role.

1D and 2D Cases: Different Diffusive Regimes

Surface effects are present in a variety of real scenarios of interest in engineering [112, 136], biological systems [137], and physics [138, 139] as a fundamental feature of several processes. For example, industrial and biochemical reactions can have the reaction rate or the sorption of reagents limited by the mass transfer between the fluid phase and the catalyst surface. In biological systems [140], the surfaces (or membranes) are responsible for the selectivity of particles by means of sorption and desorption processes, and, consequently, the particles transfer from one region to the other. Other contexts can also be found in physics such as the electrical response of water [141] or liquid crystals [142] in which the effects of the interface between electrode and fluid play an important role.