On boundary immobilization for one-dimensional Stefan-type problems with a moving boundary having initially parabolic-logarithmic behaviour

•Revisited small-time analysis for sorption in a glassy polymer.•Initial parabolic-logarithmic behaviour of the swelling front.•Self-consistent incorporation of this behaviour into a boundary immobilization scheme.•Significance for the problem of American put options in financial mathematics. In this paper, a recent one-dimensional Stefan-type model for the sorption of a finite amount of swelling solvent in a glassy polymer is revisited, with a view to formalizing the application of the boundary immobilization method to this problem. The key difficulty is that the initial behaviour of the moving boundary is parabolic-logarithmic, rather than algebraic, which has more often than not been the case in similar problems. A small-time analysis of the problem hints at how the usual boundary immobilization formalism can be recovered, and this is subsequently verified through numerical experiments. The relevance of these results to other moving boundary problems from the literature is also discussed.