DOI: 10.17587/prin.16.252-259

The Method of Integral Equations for the Stefan Problem under Low-Temperature Exposure to Biological Tissues

F. Kh. Kudayeva, Associate Professor, kfatimat@yandex.ru, Kabardino-Balkarian State University named after Kh. M. Berbekov, Institute of Artificial Intelligence and Digital Technologies, Nalchik, 360004, Russian Federation

Corresponding author: Fatimat Kh. Kudayeva, Associate Professor, Kabardino-Balkarian State University named after Kh.M. Berbekov, Institute of Artificial Intelligence and Digital Technologies, Department of Applied Mathematics and Computer Science, Nalchik, 360004, Russian Federation E-mail: kfatimat@yandex.ru

Studies of models with Stefan-type phase transitions represent a complex class of mathematical problems characterized by the presence of moving phase boundaries. In this paper, we consider a boundary value problem with Stefan-type phase transitions describing the process of low-temperature exposure to biological tissues. The methods of applying integral equations to solve such problems are analyzed, their advantages and disadvantages in comparison with other approaches are emphasized. Using the Green's function and formula, a transition is made from the Stefan problem to an equivalent system of two nonlinear integral equations of Fredholm and Volterra. The solution to the one-dimensional phase transition problem was found using the thermal potential method. Using numerical methods, computer-based solutions are obtained. The position of the moving boundary of the phase transition for a one-dimensional problem with free boundaries is determined. The results obtained will be useful to specialists involved in mathematical modeling of processes with phase transitions. They will find practical application in many fields of science and technology where a phase transition takes place.

Keywords: mathematical model, method of integral equations, numerical methods, software packages, phase transition

pp. 252—259

For citation:
Kudayeva F. Kh. The Method of Integral Equations for the Stefan Problem under Low-Temperature Exposure to Biological Tissues, Programmnaya Ingeneria, 2025, vol. 16, no. 5, pp. 252—259. DOI: 10.17587/prin.16.252-259 (in Russian).

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