Математичне моделювання процесів самозаймання у насипі з круговим перерізом методами Роте та квазіфункцій Гріна-Рвачова побудови двобічних наближень

• Дата: 2026
• Автор: Калініченко, Анатолій
• Формат: Стаття
• Мова: Українська
• Опубліковано: Kamianets-Podilskyi National Ivan Ohiienko University 2026
• Онлайн доступ: https://mcm-tech.kpnu.edu.ua/article/view/354933
• Назва журналу: Mathematical and computer modelling. Series: Technical sciences

Self-ignition of stockpiles of materials such as coal, agricultural crops, cotton, and peat is a consequence of the accumulation of heat released by an exothermic oxidation reaction; therefore, the stockpile can be considered as a body with an internal heat source. The research of self-ignition processes using mathematical modeling is reduced to the need to find a solution to the initial boundary value problem for a two-dimensional semilinear heat conduction equation. Since it is not always possible to find an analytical solution, it makes sense to use numerical analysis methods. The aim of this article is a numerical study of the initial boundary value problem for a two-dimensional semilinear heat conduction equation that arises during the mathematical modeling of self-ignition processes of a stockpile of bulk material of cylindrical shape with a circular base using Rothe's method in combination with the Green-Rvachev quasi-functions method of constructing two-sided approximations. To achieve the set goal, the original initial boundary value problem for the semilinear heat conduction equation using Rothe's method was replaced by a sequence of boundary value problems for a semilinear elliptic equation with the Helmholtz operator, each of which was reduced to the Urysohn integral equation. For this equation, an iterative process with a two-sided character of convergence and a stopping condition, which is based on an a posteriori error estimation, was constructed. Approximation of the power of the internal heat source was carried out using an exponential dependence. The results of the computational experiment are presented in the form of graphs of approximations to the solution on different time layers and graphs of heat maps, which made it possible to investigate the course of the self-ignition process of a stockpile of cylindrical shape with a circular base.