This paper deals with convergence of Picard’s successive approximations, which give a solution for perturbed body motion differential equations system through constructing a majorizing linear differential equation. The task is to determine the domain of used variables where these successive approximations converge and to estimate how large the error will be if we take a finite number of approximations. A method for finding majorant function estimates required to determine Picard’s successive approximations convergence domain is constructed.
This work discusses the problem of the convergence of Picard's successive approximations, solving the differential equation system of disturbed body motion through assembling a majorizing linear differential equation. The problem is to determine in which region of variables used these successive approximations will converge and to estimate the error, if a finite number of approximations is used.
The article is concerned with representation of the first-order disturbances in the rectangular coordinates, which are components of regularized speed and motion time of an investigated body in the form of polynomials on degrees of a certain regularizing variable, the analysis of an arrangement and nature of singular points is carried out, possibility of representation of coordinates and components of motion speed in the form of polynomials is proved. The main result of the article is the development of the method for determining the first-order disturbances in the form of finite polynomials. This representation of rectangular coordinates, of the components of regularized speed was obtained through a series of successive replacements of the independent variable. An algorithm for calculating first-order disturbances in rectangular coordinates, which are components of the speed and time of body motion.
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