Методи розв’язування початкової задачі для нелінійних інтегро-диференціальних рівнянь з оцінкою локальної похибки

• Дата: 2022
• Автори: Кунинець, Андрій, Пелех, Роман
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Kamianets-Podilskyi National Ivan Ohiienko University 2022
• Онлайн доступ: http://mcm-tech.kpnu.edu.ua/article/view/269337
• Назва журналу: Mathematical and computer modelling. Series: Technical sciences

One of the modern scientific methods of researching phenomena and processes is mathematical modeling, which in many cases allows replacing the real process and makes it possible to obtain both a qualitative and a quantitative picture of the process. Since the exact solutions of such models can be found in very individual cases, it is necessary to use approximate methods. In applied mathematics, fractional-rational approximations, which under appropriate conditions give a high rate of convergence of algorithms, bilateral and monotonic approximations have become widely used. In this work, using the technique of constructing one-step methods for solving the initial problem for ordinary differential equations and developing the sought solution into a finite continued fraction, a numerical method for solving the Cauchy problem for nonlinear integro-differential equations of the Volterra type is proposed. The values of the parameters at which the nonlinear method of the first and second order of accuracy is obtained are found. Computational formulas are proposed, which at each integration step allow obtaining an upper and lower approximation to the exact solution without additional references to the right-hand side of the integro-differential equation. Calculation formulas, in which the main terms of the local error differ only in sign, form a two-sided method. We take the half-sum of bilateral approximations to the exact solution as the approximate solution at the given integration point, and the absolute value of the half-difference determines the error of the obtained result. The modular nature of the proposed algorithms makes it possible to obtain several approximations to the exact solution of the initial problem for the nonlinear integro-differential equation at each point of integration. The comparison of these approximations gives useful information in the matter of choosing the integration step or in assessing the accuracy of the result.