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Analytical solutions for the penetration of the melting front are found by solving the motion equation of the fusion interface, obtained by the conduction heat equation for the liquid region, and by the energy conservation at the fusion front. The problem of temperature dependence of the thermal properties of the material is resolved by introducing new temperature scales; moreover, the discontinuity in the value of the thermal conductivity at the fusion interface is taken into account. Solutions are obtained both for constant and Gaussian light beams, and also for the case of constant thermal properties, while introducing the thermal conductivity discontinuity at the fusion interface. Numerical results for the specific case of Si are presented both for constant and variable thermal parameters, which compare favorably with the results obtained by computer programs.