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

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.