Exact solution for Stefan problem with general power-type latent heat using Kummer function

A one-phase Stefan problem in which the latent heat is a power function of position with a positive exponent is investigated. The Stefan problem involves a nonlinear boundary condition of the second type. The physical rational of the problem can be found in the context of both the shoreline movement and the soil freezing process. An exact solution is constructed using the similarity transformation technique and the theory of the Kummer functions. The existence and the uniqueness of the solution are proved. Computational examples of the solution provide useful data for verifying general numerical algorithms of Stefan problems. In the end, the solution for a similar Stefan problem involving a nonlinear boundary condition of the first type is also presented.