Чисельний аналіз методом двобічних наближень деяких задач стаціонарної нелінійної теплопровідності - Василишин, Костянтин; Ламтюгова, Світлана (Kamianets-Podilskyi National Ivan Ohiienko University, 2025)
The task of studying the processes of heat conduction in objects located in nonlinear environments is reduced to solving boundary value problems for a nonlinear heat conduction equation, in which the coefficients and/or the power function of heat sources vary with temperature according to certain rules. Among the numerical approaches applicable to solving such problems for nonlinear equations of mathematical physics, one can distinguish the finite difference method, finite element method, variational, projection and manifold iterative methods. Given those options, we can consider the method of two-sided approximations as the most useful since it allows the researcher to obtain a convenient estimate for the error of the approximate solution and justify the existence of a solution to the original problem. The aim of the article is to study the applicability of the method of two-sided approximations based on usage of Green's functions to solve the first boundary value problem for a nonlinear one-dimensional heat conduction equation with a power-dependent temperature coefficient of heat conduction and an exponentially temperature-dependent power function of internal heat sources. To achieve the set goal, the original problem was replaced, and the new obtained boundary value problem was reduced to the equivalent Hammerstein integral equation, which was considered as a nonlinear operator equation in a semi-ordered Banach space. The conditions for the existence of a unique positive solution to the problem and the conditions for two-sided convergence of approximations to it were formulated. The developed method was implemented programmatically and investigated when solving a test problem. The results of the computational experiment are given via graphical and tabular information