Application of the Collocation-Iterative Method to Nonlinear Integro-Functional Equations

• Дата: 2020
• Автори: Геселева, Катерина, Кріль, Сергій
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2020
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/224863
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The article investigates the application of the collocation-iterative method to one type of nonlinear integro-functional equation. The conditions that guarantee the existence of a single solution of such an equation are given. The basic algorithm of the method is given and the conditions under which this method will be convergent are indicated. In substantiating this method, the fact is used that the original nonlinear integro-functional equation can be reduced to a nonlinear integral equation, the kernels of integral operators of which are written explicitly. Another important point in the justification is that the collocation-iterative method can be interpreted as a partial case of the projection-iterative method. The application of the latter to different types of nonlinear operator equations has been systematically investigated in the works of A. Y. Luchkа and his students. A significant difference between the method studied in this work is that at each step of the iteration it is necessary to solve systems of nonlinear algebraic or transcendental equations, which is the main technical complexity of this process. But the problem of finding solutions of such systems is simpler than solving the original nonlinear integro-functional equation. The article shows that the method of successive approximations and the method of collocation of the solution of the initial nonlinear equation can be considered as partial cases of the collocation-iterative method. In addition to the basic algorithm of the method, its computational algorithm is given, which is more convenient for direct calculations than the method itself and which, if necessary, can be successfully implemented on a computer by creating an appropriate program.