Решение двухточечной задачи ЛОДУ второго порядка построением полной системы решений модифицированным методом Чебышевской коллокации

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В предыдущих работах мы разработали устойчивый быстрый численный алгоритм для решения обыкновенных дифференциальных уравнений первого порядка. Метод, основанный на чебышевской коллокации, позволяет одинаково успешно решать как начальные задачи, так и с фиксированным условием в произвольной точке интервала. Алгоритм решения краевой задачи практически реализует однопроходный аналог традиционно применяющегося в таких случаях метода стрельбы (Shooting method). В настоящей работе мы расширяем разработанный алгоритм на класс линейных ОДУ второго порядка. Активное использование метода интегрирующих множителей и метода Даламбера позволяет свести метод решения уравнений второго порядка к последовательности решений пары уравнений первого порядка. Общее решение начальной или краевой задачи для неоднородного уравнения 2-го порядка представляется в виде суммы базисных решений с неизвестными постоянными коэффициентами. Такой подход позволяет обеспечить численную устойчивость, наглядность и простоту алгоритма.

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1. Introduction The paper studies a method for solving linear ordinary differential equations (ODEs) of the second order using integrating factors [1-3]. The method of integrating factors in combination with the Chebyshev collocation method [4] was previously applied by the authors to solve first-order ODEs (of general form) [5]. Moreover, the Chebyshev collocation method was successfully applied by the authors to solve second-order linear ODEs (LODEs) using both differentiation matrices [6] and integration matrices [7]. K.P. Lovetskiy et al. developed and applied a modified Chebyshev collocation method, which turned out to be not only more reliable, but also significantly more efficient compared to previous versions of the collocation method and other Runge-Kutta-type methods (see [5-9]) or shooting method [10]. At the first stage of the two-stage modified method proposed by the authors, when expanding the approximate solution in Chebyshev polynomials (of the first or second kind), the corresponding special Gauss-Chebyshev-Lobatto grids are used, on which the search for a part of the coefficients of the approximate general solution of the ODE is reduced to solving non-degenerate and wellconditioned (with diagonal matrices) System of Linear Algebraic Equations (SLAE). At the second stage, the solution is refined by using correctly formulated initial (or boundary) conditions. In this case, the SLAE with a positive definite diagonal matrix is solved first, and then the low-dimensional (one- or two-dimensional) SLAE is solved with respect to the first coefficients of the expansion of the solution in Chebyshev polynomials. The method allows solving with equal efficiency both initial problems and problems with conditions at arbitrary points, previously solved, e.g., by the shooting method, which thus loses its relevance. Thus, we propose a constructive algorithm for approximate numerical solution of a wide class of LODEs. At the same time, the stage of the algorithm, consisting of solving the SLAE with a diagonal matrix, actually does not require computational costs, because it is reduced to a set of a small number of the simplest computational procedures. And only at the final stage, comprising the calculation of the first pair of coefficients of the expansion of the final particular solution, it is necessary to solve two-dimensional linear algebraic systems of equations determined by the initial or boundary conditions. The method has proven itself to be perfect in solving one-point problems for first-order ODEs (see [5, 8, 9]). The application of the modified Chebyshev collocation method to solving second-order ODEs has also demonstrated high efficiency. We solve two-point problems for second-order linear ODEs using [11] the two-stage Chebyshev collocation method. The first stage is devoted to finding an approximate solution to the ODE in the form of a Chebyshev polynomial [12] with undetermined first coefficients. At the second stage, the first coefficients (if they exist) are found by solving a 2 × 2 SLAE [5-7, 13]. The first stage can be implemented in several not entirely equivalent ways [14]. Ref. [6] presents the Chebyshev collocation method for obtaining a solution to a second order LODE using the Chebyshev differentiation matrix [15]. The paper [7] implements the Chebyshev collocation method for obtaining a solution to a second order LODE using the Chebyshev antidifferentiation matrix. The authors noted that constructing a general (complete) solution from the individual partial solutions of the LODE obtained in this way seems to be a computationally complex task. At the same time, using an intermediate method that makes use of integrating factors to reduce the LODE to the form of a total derivative allows one to obtain general (complete) solutions of the second-order LODE more efficiently. In the present article we seek approximate solutions of linear second-order ODEs of a rather general form
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Об авторах

К. П. Ловецкий

Российский университет дружбы народов

Email: lovetskiy-kp@rudn.ru
ORCID iD: 0000-0002-3645-1060

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Computational Mathematics and Artificial Intelligence

ул. Миклухо-Маклая, д. 6, Москва, 117198, Российская Федерация

М. Д. Малых

Российский университет дружбы народов; Объединённый институт ядерных исследований

Email: malykh-md@rudn.ru
ORCID iD: 0000-0001-6541-6603
Scopus Author ID: 6602318510
ResearcherId: P-8123-2016

Doctor of Physical and Mathematical Sciences, Head of the Department of Computational Mathematics and Artificial Intelligence of RUDN University

ул. Миклухо-Маклая, д. 6, Москва, 117198, Российская Федерация; ул. Жолио-Кюри, д. 6, Дубна, 141980, Российская Федерация

Л. А. Севастьянов

Российский университет дружбы народов; Объединённый институт ядерных исследований

Email: sevastianov-la@rudn.ru
ORCID iD: 0000-0002-1856-4643

Professor, Doctor of Sciences in Physics and Mathematics, Professor at the Department of Computational Mathematics and Artificial Intelligence of RUDN University, Leading Researcher of Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research

ул. Миклухо-Маклая, д. 6, Москва, 117198, Российская Федерация; ул. Жолио-Кюри, д. 6, Дубна, 141980, Российская Федерация

С. В. Сергеев

Российский университет дружбы народов

Автор, ответственный за переписку.
Email: 1142220124@rudn.ru
ORCID iD: 0009-0004-1159-4745

PhD student of Department of Computational Mathematics and Artificial Intelligence

ул. Миклухо-Маклая, д. 6, Москва, 117198, Российская Федерация

Список литературы

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© Ловецкий К.П., Малых М.Д., Севастьянов Л.А., Сергеев С.В., 2024

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