Diffusion and decay in two dissimilar half-spaces in contact

We formulate the problem of diffusion-decay processes occurring in two half-spaces separated by a boundary (a one-dimensional problem). The equations are solved by Laplace transforms, by finite differences, and by a Greens-function technique. The inverse Laplace transform integrals are evaluated numerically and agree with the finite-difference calculations. Thus each may be regarded as accurate. Late-time asymptotic approximations are derived from the Laplace representation, and early-time approximations are derived from the Greens-function representation. These are compared with the numerical integration and finite-difference results. We also introduce a method for modeling the processes by random walks, which treats the complications introduced by the presence of the boundary. All of the above-noted calculational methods are compared. It is especially important that the asymptotic approximations are accurate in their domain of validity, as they allow us to introduce a physical picture regarding the competition of diffusive and decay processes.