New algorithmic implementation of exact three-point difference schemes for systems of nonlinear ordinary differential equations of the second ord - Kutniv, M. V.; Krol М. (Ukrains’kyi Matematychnyi Zhurnal, 2022)
UDC 519.62 In this paper, we propose and justify three-point difference schemes of higher order of accuracy on a non-uniform grid for systems of nonlinear ordinary differential equations of the second order with a derivative on the right-hand side and boundary conditions of the first kind. We construct a new approximation of the derivative of the solution to the boundary value problem at grid nodes, prove the existence and uniqueness of the solution, and establish the order of accuracy of the difference schemes. Additionally, an iterative Newton-type method for determining the solution of these schemes is developed and an algorithm for the automatic selection of grid points, which guarantees the achievement of the specified accuracy, is suggested. We also present numerical solutions of some examples, which confirm the efficiency and reliability of our algorithm.