Analysis of One-Dimensional Steady-State Thermochemical Problems Using the Method of Two-Sided Approximations with Green’s Function

• Date: 2026
• Main Author: Янбеков, Равіль
• Format: Article
• Language: Ukrainian
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2026
• Online Access: https://mcm-math.kpnu.edu.ua/article/view/361386
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

This paper considers the first boundary value problem for a semilinear ordinary differential equation, which serves as a mathematical model of a thermochemical process. In particular, the classical Bratu problem and two of its generalizations are studied, taking into account both heat losses due to cooling and external heating. The exponential nonlinearity in the equations corresponds to the Frank-Kamenetskii approximation of the Arrhenius law. Using Green’s function, each of the considered problems is reduced to an equivalent Hammerstein integral equation, which is analyzed within the framework of nonlinear operator theory in semi-ordered Banach spaces. For this purpose, the integral equation is represented as an operator equation with a nonlinear operator acting in the space of continuous functions partially ordered by the cone of nonnegative functions. The operator is investigated with respect to positivity, monotonicity, Lipschitz continuity, and the existence of an invariant conical segment. For the numerical analysis of these integral equations (and hence the corresponding boundary value problems), iterative schemes based on the method of two-sided approximations are proposed. The endpoints of the invariant conical segment are chosen as initial approximations. For each scheme, convergence conditions and conditions for the existence of positive solutions to the corresponding boundary value problems are established. Additionally, two-sided a priori estimates for these solutions are obtained. Computational experiments are carried out for various parameter values, and in the case of the Bratu problem, the results are compared with the exact solution. Based on the analysis, conclusions are drawn regarding the efficiency of the proposed computational schemes. In particular, their advantages include the availability of guaranteed a posteriori error estimates for the approximate solution and a convenient stopping criterion for the iterative process. The results obtained in this work can be extended to two- and three-dimensional mathematical models of thermochemical processes with exponential nonlinearities.