Application of the Method of Two-Sided Approximations for Finding Positive Axially Symmetric Solutions of Boundary Value Problems with Singular Nonlinearities

In this paper, we consider the problem of finding positive axially symmetric solutions to boundary value problems for nonlinear elliptic differential equations using the method of two-sided approximations. The first boundary value problem, or Dirichlet problem, is solved. The nonlinearity involved is of an anti-monotonic type, characterized by a power-law dependence with an exponent ranging from –1 to 0. By transitioning to a polar coordinate system and exploiting the axial symmetry of the solution, the original boundary value problem for an elliptic equation is reduced to a boundary value problem for an ordinary differential equation on a finite interval. The solution depends solely on the radial coordinate, eliminating the dependence on the angular variable. In this case, the pole of the polar coordinate system becomes a singular point, where it is necessary to impose a boundedness condition on the solution. For the boundary value problem, the Green’s function is constructed, after which the problem is reduced to a Hammerstein integral equation. This integral equation is treated as a nonlinear operator equation in a Banach space of continuous functions on a finite interval, partially ordered by the cone of non-negative functions on that interval. The corresponding operator is studied with respect to properties such as anti-monotonicity (antitonicity), positivity, boundedness, and pseudo-concavity. The next stage involves determining the initial approximation as the endpoints of a strongly invariant conical segment for the antitone operator, in a way that ensures the highest possible convergence rate of the iterative process. Two iterative sequences of two-sided approximations are then constructed. The first sequence is non-decreasing with respect to the cone, while the second is non-increasing. At each iteration, the arithmetic mean of the upper and lower approximations is chosen as the current estimate. The iterative process continues until the error estimate for the solution meets the prescribed accuracy. The theoretical results obtained in this work were verified through computational experiments. The dependence of the solution and the convergence rate of the iterative process on the parameters in the right-hand side was analyzed and illustrated with corresponding graphs.