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We consider subdiffusion in a system which consists of two media separated bya thin membrane. The subdiffusion parameters may be different in each of themedium. Using the new method presented in this paper we derive theprobabilities (the Green's functions) describing a particle's random walk inthe system. Within this method we firstly consider the particle's random walkin a system with both discrete time and space variables in which a particle canvanish due to reactions with constant probabilities $R_1$ and $R_2$, definedseparately for each medium. Then, we move from discrete to continuousvariables. The reactions included in the model play a supporting role. We linkthe reaction probabilities with the other subdiffusion parameters whichcharacterize the media by means of the formulae presented in this paper.Calculating the generating functions for the difference equations describingthe random walk in the composite membrane system with reactions, which dependexplicitly on $R_1$ and $R_2$, we are able to correctly incorporate thesubdiffusion 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 functionsfor the composite membrane system without any reactions. From the obtainedGreen's functions, we derive the boundary conditions at the thin membrane. Oneof the boundary conditions contains the Riemann--Liouville fractional timederivative, which shows that the additional `memory effect' is created in thesystem. As is discussed in this paper, the `memory effect' can be created bothby the membrane and by the discontinuity of the medium at the point at whichthe various media are joined.

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