Analysis By the Method of Two-Sided Approximations of Positive Axially Symmetric Solutions of the First Boundary Value Problem for the Helmholtz Equation with a Monotone Power Nonlinearity

• Date: 2025
• Main Authors: Пархоменко, Владислав, Сидоров, Максим
• Format: Article
• Language: Ukrainian
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2025
• Online Access: http://mcm-math.kpnu.edu.ua/article/view/346580
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

The paper presents an analysis, using the method of two-sided approximations, of positive axially symmetric solutions to the first boundary value problem for a semilinear elliptic differential equation with the Helmholtz operator. The domain in which the problem is considered is a disk, on the boundary of which a homogeneous first boundary condition is imposed. The nonlinearity is monotone and is described by a power dependence on the unknown function, where the exponent varies from 0 to 1. By passing to polar coordinates and taking into account that the solution is axially symmetric (that is, it does not depend on the angular variable and depends only on the distance from the center of the disk), we obtain a boundary value problem for a semilinear ordinary differential equation. The pole of the polar coordinate system is a singular point of this equation, which necessitates imposing a boundedness condition on the solution at this point. For the boundary value problem, a Green’s function is constructed and a transition is made to an equivalent Hammerstein integral equation, which is considered as a nonlinear operator equation in a Banach space of functions continuous on an interval, semi-ordered by the cone of nonnegative functions on this interval. The properties of the corresponding integral operator, such as monotonicity (isotonicity), positivity, boundedness, and concavity are investigated. At the next stage of the study, the endpoints of an invariant conical segment are determined, which serve as initial approximations for the iterative process. After that, two parallel iterative processes are constructed. The first iterative sequence is nondecreasing with respect to the cone (lower approximations), while the second is nonincreasing with respect to the cone (upper approximations). At each iteration, the current approximation is chosen as the arithmetic mean of the upper and lower approximations. Thus, at each step, the iterative process provides an a posteriori error estimate. As a result, the existence and uniqueness of a positive axially symmetric solution to the considered problem are established. The theoretical results obtained in the paper were confirmed by conducting a computational experiment. The dependence of the solution and the convergence rate of the iterative process on the parameters of the equation were analyzed and illustrated by the corresponding graphs