Model of anomalous diffusion--absorption process in a system consisting of two different media separated by a thin membrane

TL;DR

This paper develops a mathematical model for anomalous diffusion and absorption in a two-media system separated by a semi-permeable membrane, deriving boundary conditions and solutions applicable to various diffusion types.

Contribution

It introduces a new model with boundary conditions for anomalous diffusion across a membrane, including subdiffusion and superdiffusion, using Laplace transform techniques.

Findings

Derived Green's functions for the system

Established boundary conditions at the membrane

Provided methods for inverse Laplace transform calculations

Abstract

We present the model of a diffusion-absorption process in a system which consists of two media separated by a thin partially permeable membrane. The kind of diffusion as well as the parameters of the process may be different in both media. Based on a simply model of particle's random walk in a membrane system we derive the Green's functions, then we find the boundary conditions at the membrane. One of the boundary conditions are rather complicated and takes a relatively simple form in terms of the Laplace transform. Assuming that particles diffuse independently of one another, the obtained boundary conditions can be used to solve to differential or differential-integral equations describing the processes in multilayered systems for any initial condition. We consider normal diffusion, subdiffusion and slow subdiffusion processes, and we also suggest how superdiffusion could be included in this model. The presented method provides the functions in terms of the Laplace transform and some useful methods of calculation the inverse Laplace transform are shown.