Modification of the Basic Two-Sided Method for Solving Integral Equations

• Date: 2026
• Main Authors: Сеньо, Петро, Заяць, Артур
• Format: Article
• Language: Ukrainian
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2026
• Online Access: https://mcm-math.kpnu.edu.ua/article/view/360580
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

The research considers the problem of constructing guaranteed two-sided approximations of the solution of an integral equation of a certain kind. Classical methods typically yield only approximation of the solution at the certain points, but at other points there are general theoretical approximations without rigorous error bounds, whereas for problems with uncertainty in parameters or initial conditions it is essentially important to obtain upper and lower bounds that definitely contain the sought solution. Classical interval methods for solving such equations based on interval analogues of the Taylor series require automatic differentiation of high orders and are characterized by the accumulation of error on large intervals, that significantly complicates their practical application. In [1] it presents an iterative two-sided algorithm for solving equations of aforesaid form based on the mathematics of functional intervals with quadratic convergence, in which, however, the choice of the interval length that provides the given accuracy and the number of iterations of narrowing the linear functional interval of the solution are determined implicitly. In this paper a theorem is proved that yields an estimate for the functional uncertainty of the two-sided approximation at the right boundary of the interval, an explicit expression of the partition step ensuring the prescribed accuracy of the approximation in a single construction, and a logarithmic estimate of the number of iterations of narrowing of the linear functional interval of the solution on given interval by the basic algorithm of [1]. Based on this proven theorem, a modified algorithm is proposed that provides the desired narrowing of the two-sided approximation of the desired solution in one step. The algorithm is tested on two numerical experiments, which results confirm the theoretical conclusions. This substantially reduces the number of algorithm iterations.