Exact and τ-method approximate solutions to a two-phase Stefan problem with Robin condition and exponential sources

Abstract

This work examines a two-phase Stefan problem in a semi-infinite domain with a convective (Robin-type) boundary condition at the fixed face and with exponential internal heat sources in terms of a similarity-type variable. Existence and uniqueness of the similarity-type solution are established with exactness, and asymptotic analysis of the system is carried out, showing convergence towards the associated two-phase Stefan problem with the imposed temperature condition at the fixed face in the limit when the heat transfer coefficient tends to infinity. A numerical example pertaining to melting paraffin is given to support the theoretical results. Moreover, the Tau-method based on shifted Chebyshev and Legendre polynomials are employed to construct approximate solutions that are then compared to the exact similarity-type solution that serves as benchmark for quantifying the accuracy of spectral approximations.