Stefan problem

In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem (also Stefan task) is a particular kind of boundary value problem for a partial differential equation (PDE), adapted to the case in which a phase boundary can move with time. The classical Stefan problem aims to describe the temperature distribution in a homogeneous medium undergoing a phase change, for example ice passing to water: this is accomplished by solving the heat equation imposing the initial temperature distribution on the whole medium, and a particular boundary condition, the Stefan condition, on the evolving boundary between its two phases. Note that this evolving boundary is an unknown (hyper-)surface: hence, Stefan problems are examples of free boundary problems.

The problem is named after Josef Stefan (Jožef Stefan), the Slovenian physicist who introduced the general class of such problems around 1890, in relation to problems of ice formation. This question had been considered earlier, in 1831, by Lamé and Clapeyron.

From a mathematical point of view, the phases are merely regions in which the solutions of the underlying PDE are continuous and differentiable up to the order of the PDE. In physical problems such solutions represent properties of the medium for each phase. The moving boundaries (or interfaces) are infinitesimally thin surfaces that separate adjacent phases; therefore, the solutions of the underlying PDE and its derivatives may suffer discontinuities across interfaces.

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