Applіcatіon of the Mathematіcs of Functіonal Іntervals to Solvіng Some Types of General Іnіtіal Problems - Сеньо, Петро (Кам'янець-Подільський національний університет імені Івана Огієнка, 2026)
This research provides a method for solving the general initial problem and iterative algorithms for constructing two-sided approximations based on the mathematics of functional intervals for solving the Cauchy initial problem for ordinary differential equations and some types of integral equations. Solutions of such problems are obtained in the form of functional intervals. An adaptive choice of the step length in these algorithms automatically responds to the problem nonlinearity degree. It is proved that functional equations, integral equations, and the Cauchy problem are special cases of the general initial problem. Based on the concept of a function, its derivative, and the tangent, the basic development principle of the functional dependence is proposed, the consistency principle of the direct and reverse development of the functional dependence from opposite ends of the integration interval of the functional interval of the function and the functional intervals of its derivatives. It is shown that except for the tabular and analytical representation, the functional dependence can be presented in the tabular-analytical form. The lemmas and theorems proved in this research make it possible to analyze and eliminate various uncertainties associated with the continuously differentiable functions. In this research there is the method proposed for solving the general initial problem and iterative algorithms for constructing two-sided approximations based on the mathematics of functional intervals that combines the idea of two-sided approximations with the mathematical apparatus of functional intervals. The adaptive choice of the step length in these algorithms automatically responds to the level of nonlinearity of the problem, and the quadratic convergence of the width of the functional interval of the problem solution provides an effective narrowing of the uncertainty of the solution at each iteration.