The Stefan problem of a semi-infinite material with arbitrarily prescribed initial and flux conditions is studied. When the surface temperature is initially different from the freezing temperature, there exists a presolidification or a premelting period prior to the occurrence of a phase change. The exact solutions for both periods, before and after the appearance of a phase change, are established. This indicates that the initial condition of the Stefan problem with a prescribed heat flux at the surface cannot be assumed to be constant. Possibilities of similarity solutions of the problem are also examined. A similarity solution exists only when the heat flux is proportional to t−1/2 and the initial and boundary conditions satisfy an inequality. The solidification of a supercooled liquid is also investigated. The exact solution is obtained.

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