Subdiffusion--absorption process in a system consisting of two different media

TL;DR

This paper models subdiffusion with reaction in a two-media system, deriving solutions and boundary conditions that account for different subdiffusion parameters and a reactive process, with particles crossing media boundaries freely.

Contribution

It introduces a method to derive Green's functions and boundary conditions for subdiffusion-reaction systems with different parameters in two media.

Findings

Derived fundamental solutions (Green's functions) for the system.

Established boundary conditions at the media interface.

Demonstrated the continuity of flux and fractional derivative boundary condition.

Abstract

Subdiffusion with reaction $A+B\rightarrow B$ is considered in a system which consists of two homogeneous media joined together; the $A$ particles are mobile whereas $B$ are static. Subdiffusion and reaction parameters, which are assumed to be independent of time and space variable, can be different in both media. Particles $A$ move freely across the border between the media. In each part of the system the process is described by the subdiffusion--reaction equations with fractional time derivative. By means of the method presented in this paper we derive both the fundamental solutions (the Green's functions) $P(x,t)$ to the subdiffusion--reaction equations and the boundary conditions at the border between the media. One of the conditions demands the continuity of a flux and the other one contains the Riemann--Liouville fractional time derivatives $\partial^{\alpha_1}P(0^+,t)/\partial t^{\alpha_1}=(D_1/D_2)\partial^{\alpha_2}P(0^-,t)/\partial t^{\alpha_2}$, where the subdiffusion parameters $\alpha_1$, $D_1$ and $\alpha_2$, $D_2$ are defined in the regions $x<0$ and $x>0$, respectively.