Substantiation of the Averaging Method for a Nonlocal m-Frequency Problem with Linearly Transformed Arguments

• Дата: 2020
• Автори: Скутар, Ігор, Бігун, Ярослав
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2020
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/224957
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The system of differential equations with delay on a finite interval with slow and fast variables is investigated. The delay in the system is characterized by linearly transformed arguments in slow and fast variables. Integral conditions are given for slow and fast variables. A characteristic feature of such systems is the appearance of resonances in the process of evolution. The condition for the resonance in the system contains a dependence on the delays in fast variables. An effective method for the research of multi-frequency systems is the method of averaging, the substantiation of which for systems without delay of the argument is obtained in the works of V. I. Arnold, E. O. Grebenikov, M. M. Khapaiev, А. М. Samoilenko, R. I. Petryshyn. This paper uses the method proposed by A. M. Samoilenko which is based on the estimation of oscillating integrals. In this paper, the procedure of averaging over fast variables is carried out both in the system of equations and in integral conditions. In the average problem, the variables are separated and the problem for slow variables is solved independently of the fast variables. Finding fast variables is reduced to the problem of integration. The existence of a unique solution for the problem in the class of continuously differentiated functions is proved. The accuracy estimation of the averaging method is obtained, which clearly depends on the small parameter and the number of fast variables and linearly transformed arguments in them. An estimate of the value of the small parameter was also found. The condition of passing resonant zones is reduced to checking the difference from zero of the Vronsky determinant, constructed by the frequency system taking into account the number of linearly transformed arguments. An example of a single-frequency system with integral conditions is constructed, on which the obtained result is illustrated; accuracy estimation and values of the small parameter are obtained.