Karolina Pawlak
supervisor: Ewa Zadrzyńska-Piętka, Adam Kubica
The fractional Stefan problem is a free-boundary problem of evolution type. It describes a change of phase in medium e.g. melting of solid. In the classical formulation there exists an interphase which separates the solid and liquid parts.
During the process of melting the interphase is moving and we have two unknowns: the temperature and the function describing the interface.
We focus on the one dimensional problem. Moreover, we consider the one phase problem i.e. we assume that the temperature in solid part is equal to zero. The mathematical formulation is based on the principle of energy conservation, where the flux is given by the Riemann-Liouville fractional derivative of temperature gradient. Due to this form of the flux, the model exhibits memory effects. It means that the total enthalpy in the domain at time t is a sum of the initial enthalpy and the time-average of differences of local fluxes at the endpoints of the domain.
During the presentation, I will briefly present the definitions of the fractional operators and I would like to discuss the derivation of the fractional Stefan problem. Furthermore, I am going to introduce the weak formulation of the one dimensional, one-phase fractional Stefan problem. In the weak Stefan problem there is only one unknown: the enthalpy. We assume that the enthalpy jumps at the change of phase. As a result, the transformation process from solid to liquid occurs through the mushy region and no interphase is formed.