Modeling of Boundary Value Problems for Linear Neutral Delay Differential-Difference Equations

• Дата: 2020
• Автори: Черевко, Ігор, Дорош, Андрій, Перцов, Андрій, Гаюк, Іван
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2020
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/224963
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

In space navigation problems, optimal control of systems with aftereffect, ecology and immunology problems, boundary value problems for differential-difference and integro-differential equations with delays arise, which are an important part of modern theory of differential-functional equations. Finding precise solutions of boundary value problems for differential-difference equations is possible only for the simplest classes of such problems. At present, projection-iterative methods, numerical-analytical method and others are suggested for boundary value problems with delay and of neutral type. The spline-collocation method for solving boundary value problems for differential-difference equations is one of the most efficient algorithms that allows building simple computational schemes. In this paper, we investigate the scheme of modeling boundary value problems for linear differential-difference equations of neutral type with many variable deviations of the argument. A functional space is defined to which the solutions of the considered boundary value problems belong, the properties of the solution smoothness are investigated depending on the structure of the argument deviations. Simple and verifiable sufficient conditions for the boundary value problem solution existence are given. For finding the solution of the boundary value problem, an iterative computational scheme based on the spline approximation method is described. In order to take into account possible discontinuities of the boundary value problem solution derivatives, cubic splines of defect two are used for neutral-type equations. Coefficient conditions for the initial equation which ensure the convergence of the iterative process are obtained. An estimate of the approximate solution error is conducted. A model example of a boundary value problem for a neutral type differential-difference equation is presented on which the iterative scheme is demonstrated. Numerical experiments confirm the obtained theoretical results.