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

T. Nauryz^abc, S. N. Kharin^b, A. Briozzo^d, J. Bollati<sup>d

a</sup> School of Digital Technologies, Narxoz University, 55 Zhandosov St, 050035 Almaty, Republic of Kazakhstan
^b Department of Mathematical Physics and Modelling, Institute of Mathematics and Mathematical Modelling, 125 Pushkin St, 050010 Almaty, Republic of Kazakhstan
^c International School of Economics, Kazakh-British Technical University, 59 Tole Bi St, 050005 Almaty, Republic of Kazakhstan
^d Department of Matematics, Universidad Austral, CONICET, 1950 Paraguay St, S2000FZF, Rosario, Argentina

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 satis es a generalized heat equation with the Thomson effeect. 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.

Keywords and phrases: Stefan problem, generalized heat equation, Thomson effect, similarity solution, non-linear integral equations, nonlinear thermal coefficient, fixed point theorem.

MSC: 80A22, 80A05, 35C11

Received: 28.08.2024

Language: English

DOI: 10.32523/2077-9879-2025-16-3-68-89