The Clocation-Iterative Method of Solving Integro-Functional Equations with Restrictions

• Date: 2018
• Main Author: Геселева, Катерина Григорівна
• Format: Article
• Language: Ukrainian
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2018
• Online Access: http://mcm-math.kpnu.edu.ua/article/view/159383
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

Integro-functional equations are widely used in various fields of science and science (in particular, to such equations with a deviation of the argument as a neutral type and with a delay).In some cases, solving these equations is known additional information. Therefore, it is important not only the solution of this equation but also the establishment of compatibility conditions of the corresponding problem, that is, one needs to find out if the desired solution of the problem with the additional conditions is consistent.A set of scientific papers is devoted to the establishment of compatibility conditions of problems of this type with respect to different types of operator equations and the development of methods for constructing their solutions [1–4].The article deals with one type of integro-functional equation with condition and restrictions on the desired function, which are integral in nature. The terms of the compatibility of the original problem are formulated. With regard to the values included in the given task, it is required that they satisfy a number of necessary conditions. It is shown that under these conditions the initial problem will be equivalent to some Fredholm integral equation of a second kind with a completely continuous operator and additional conditions on the desired solution.In addition to the main task also considered ancillary task — a task with management, when in the case of compatibility, an additional, adjusting value is introduced. The terms of compatibility of the original problem are formulated and substantiated.In the paper, the iterative and colocation-iterative methods of constructing approximate solutions of the original problem with constraints are also presented and grounded. The algorithm of these methods and sufficient conditions of their convergence are indicated. At the same time, we use the fact that the initial problem in the fulfillment of certain conditions is equivalent to an integral equation with constraints. The presented methods of constructing approximate solutions of the integro-functional equation with additional conditions can be successfully implemented on the computer by creating the corresponding programs.