Reaction-diffusion equations describe a number of physical, chemical, and biological phenomena, many of which occur in composite environments with piece-wise constant diffusion coefficients. We develop semi-analytical solutions of axisymmetric reaction-diffusion equations with first-order reaction kinetics and continuous transient boundary conditions. These solutions are directly applicable to heat conduction in composite media with transient boundary conditions and heat generation. The solutions lose their robustness in the long time regime, when the Laplace variable tends to zero. This limitation is overcome by the use of corresponding steady-state solutions.