Explicit Solution for the one-phase Stefan problem with latent heat depending on the position and a convective boundary condition at the fixed face

Abstract

An explicit solution of a similarity type is obtained for a one-phase Stefan problem in a semi-infinite material using Kummer functions. Itis considered a phase-change problem with a latent heat defined as a powerfunction of the position with a non-negative real exponent and a convectiveboundary condition at the fixed face x = 0. Existence and uniqueness of thesolution is proved. Relationship between this problem and the problems withtemperature and flux boundary condition is also analysed. Furthermore it isstudied the limit behaviour of the solution when the coefficient which char-acterizes the heat transfer at the fixed boundary tends to infinity. Comput-ing this limit allows to demonstrate that the problem proposed in this paperwith a convective boundary condition generalizes the problem with Dirichletboundary condition. Numerical computation of the solution is done over cer-tain examples, with a view to comparing this results with those obtained bygeneral algorithms that solve Stefan problems.