Subdiffusion in a system consisting of two different media separated by a thin membrane

TL;DR

This paper develops a new method to derive Green's functions for subdiffusion in a two-media system separated by a thin membrane, incorporating reactions and memory effects via fractional derivatives.

Contribution

The paper introduces a novel approach linking discrete random walk models with continuous subdiffusion equations, including reactions, and derives boundary conditions with fractional derivatives.

Findings

Green's functions for the system with and without reactions

Boundary conditions involving fractional derivatives

Memory effects induced by membrane and medium discontinuity

Abstract

We consider subdiffusion in a system which consists of two media separated by a thin membrane. The subdiffusion parameters may be different in each of the medium. Using the new method presented in this paper we derive the probabilities (the Green's functions) describing a particle's random walk in the system. Within this method we firstly consider the particle's random walk in a system with both discrete time and space variables in which a particle can vanish due to reactions with constant probabilities $R_1$ and $R_2$, defined separately for each medium. Then, we move from discrete to continuous variables. The reactions included in the model play a supporting role. We link the reaction probabilities with the other subdiffusion parameters which characterize the media by means of the formulae presented in this paper. Calculating the generating functions for the difference equations describing the random walk in the composite membrane system with reactions, which depend explicitly on $R_1$ and $R_2$, we are able to correctly incorporate the subdiffusion parameters of both the media into the Green's functions. Finally, placing $R_1=R_2=0$ into the obtained functions we get the Green's functions for the composite membrane system without any reactions. From the obtained Green's functions, we derive the boundary conditions at the thin membrane. One of the boundary conditions contains the Riemann--Liouville fractional time derivative, which shows that the additional `memory effect' is created in the system. As is discussed in this paper, the `memory effect' can be created both by the membrane and by the discontinuity of the medium at the point at which the various media are joined.