Exact solution to a Stefan-type problem for a generalized heat equation with the Thomson effect

Abstract

We study a one-dimensional Stefan type problem which models the behavior of electromagnetic fields and heat transfer in closed electrical contacts that arises, when an instantaneous explosion of the micro-asperity occurs. This model involves vaporization, liquid and solid zones, in which the temperature satisfies a generalized heat equation with the Thomson effect. Accounting for the nonlinear thermal coefficient, the model also incorporates temperature-dependent electrical conductivity. By employing a similarity transformation, the Stefan-type problem is reduced to a system of coupled nonlinear integral equations. The existence of a solution is established using the fixed point theory in Banach spaces.