  We develop a new semi-analytical method for solving multilayer diffusion
problems with time-varying external boundary conditions and general internal
boundary conditions at the interfaces between adjacent layers. The convergence
rate of the semi-analytical method, relative to the number of eigenvalues, is
investigated and the effect of varying the interface conditions on the solution
behaviour is explored. Numerical experiments demonstrate that solutions can be
computed using the new semi-analytical method that are more accurate and more
efficient than the unified transform method of Sheils [Appl. Math. Model.,
46:450-464, 2017]. Furthermore, unlike classical analytical solutions and the
unified transform method, only the new semi-analytical method is able to
correctly treat problems with both time-varying external boundary conditions
and a large number of layers. The paper is concluded by replicating solutions
to several important industrial, environmental and biological applications
previously reported in the literature, demonstrating the wide applicability of
the work.
