Коллокационные алгоритмы решения уравнений Вольтерры

• Дата: 2018
• Автори: Дячук, Олександр Анатолійович, Костьян, Наталія Леонідівна
• Формат: Стаття
• Мова: rus
• Опубліковано: Kamianets-Podilskyi National Ivan Ohiienko University 2018
• Онлайн доступ: http://mcm-tech.kpnu.edu.ua/article/view/140024
• Назва журналу: Mathematical and computer modelling. Series: Technical sciences

Despite the extensive use of the collocation method for solving integral equations with constant integration limits, little attention has been paid so far to the implementation of this method with respect to integral equations with variable limits. In this article tasks of solving Volterra integral equations of 1 and 2 kinds were considered. An approximate solution is defined as a piecewise-smooth polynomial composed of polynomials over sections of the domain of definition of the variable of integration. The algorithm of the method is an iterative process. The problem is reduced to solving systems in the general case of non-linear equations with respect to the coefficients of the corresponding polynomials. At each step of the iteration, an analytic expression for the next polynomial is determined, which allows finding a solution at any point of the given interval. A special feature of the collocation algorithm for Volterra equations of the 2nd kind is the replacement of integrals by quadrature formulas, which are comprised into the system of equations with respect to the approximate values of the coefficients. The choice of the coefficients of quadrature formulas depends on the accepted number of nodes in the section. A special case of a system for three nodes is considered in the article. In doing so, the integrand of the solved equation was replaced by an interpolation polynomial in the Newton form. The results of the solution of the test cases confirm the efficiency of the proposed algorithms and indicate the high accuracy of the calculations. The collocation method allows to obtain solutions of the Volterra equations for the segments of the integration interval, choosing their length and applying on each of them an approximating expression with a small number of coordinate functions. This method can be used in identifying of the dynamic objects and systems, as well as in solving problems of reduction input signals.