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 Singular Power Nonlinearity

• Дата: 2026
• Автор: Пархоменко, Владислав
• Формат: Стаття
• Мова: Українська
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2026
• Онлайн доступ: https://mcm-math.kpnu.edu.ua/article/view/354815
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The paper analyzes, by means of the method of two-sided approximations, positive axially symmetric solutions of the first boundary value problem for a semilinear elliptic differential equation with the Helmholtz operator and a singular power nonlinearity. The problem is considered in a circular domain with a homogeneous Dirichlet condition on the boundary. The nonlinearity has an antimonotonic character and is described by a power dependence, where the exponent takes values from −1 to 0. By transforming to polar coordinates and taking into account that the solution is axially symmetric (that is, there is no dependence on the rotation angle and only the dependence on the distance from the center of the circle remains), a boundary value problem for a semilinear ordinary differential equation is obtained. In this case, 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 problem under consideration, the Green's function is constructed, followed by a reduction to an equivalent Hammerstein integral equation, which is treated as a nonlinear operator equation in a Banach space of functions continuous on a segment and semi-ordered by the cone of nonnegative functions on this segment. The properties of the corresponding integral operator, such as antimonotonicity (antitonicity), positivity, boundedness, and pseudoconcavity, are investigated. The next stage of the study involves determining the endpoints of a strongly invariant conical segment, 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 (a sequence of lower approximations), while the second is nonincreasing with respect to the cone (a sequence of upper approximations). At each iteration, the arithmetic mean of the upper and lower approximations is chosen as the current approximation. In this way, an a posteriori error estimate is obtained at every step of the iterative process. A conclusion is drawn about the existence and uniqueness of a positive axially symmetric solution to the problem under consideration. 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 the corresponding results are presented in the relevant graphs.