Analysis by the Method of Two-Sided Approximations of a Stationary Reaction–Diffusion Model in a Spherical Pellet with Arrhenius Kinetics

• Datum: 2026
• 1. Verfasser: Пархоменко, Владислав
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Кам'янець-Подільський національний університет імені Івана Огієнка 2026
• Online Zugang: https://mcm-math.kpnu.edu.ua/article/view/361414
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The paper analyzes, by means of the method of two-sided approximations, a stationary reaction–diffusion model in a spherical granule with Arrhenius kinetics. The problem is considered in a spherical domain with a nonhomogeneous Dirichlet boundary condition on the boundary, which is transformed into a homogeneous one after an appropriate substitution. The nonlinearity is represented as the product of a linear and an exponential function. After transforming to the spherical coordinate system and taking radial symmetry into account (the solution depends only on the distance from the center of the sphere and does not depend on the angular variables), the problem is reduced to a boundary value problem for a semilinear ordinary differential equation. Since the pole of the spherical coordinate system is a singular point of the obtained equation, it is necessary to impose a boundedness condition on the solution at this point. For the problem under consideration, the Green’s function is constructed, after which the problem is reduced to an equivalent integral equation, which is treated as a nonlinear operator equation in the Banach space of functions continuous on a segment and semiordered by the cone of nonnegative functions on this segment. The properties of the corresponding integral operator, such as heterotonicity and positivity, are investigated. Next, the endpoints of a strongly invariant conical segment are determined, serving as initial approximations for the iterative process. Then, two iterative processes are constructed. The first iterative sequence is nondecreasing with respect to the cone (the sequence of lower approximations), while the second one is nonincreasing with respect to the cone (the sequence of upper approximations). At each iteration step, the current approximation is chosen as the arithmetic mean of the upper and lower approximations, which makes it possible to obtain an a posteriori error estimate at every step of the iterative process. As a result, the existence and uniqueness of a positive radially symmetric solution to the considered problem are established. The theoretical results obtained in the paper were confirmed by means of a computational experiment. The results of the computational experiment are presented graphically.