Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)4367010.22363/2658-4670-2024-32-4-414-424DHGEBYModeling and SimulationМатематическое моделированиеResearch ArticleSolving a two-point second-order LODE problem by constructing a complete system of solutions using a modified Chebyshev collocation methodРешение двухточечной задачи ЛОДУ второго порядка построением полной системы решений модифицированным методом Чебышевской коллокацииhttps://orcid.org/0000-0002-3645-1060LovetskiyKonstantin P.ЛовецкийК. П.

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

lovetskiy-kp@rudn.ruhttps://orcid.org/0000-0001-6541-66036602318510P-8123-2016MalykhMikhail D.МалыхМ. Д.

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

malykh-md@rudn.ruhttps://orcid.org/0000-0002-1856-4643SevastianovLeonid A.СевастьяновЛ. А.

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

sevastianov-la@rudn.ruhttps://orcid.org/0009-0004-1159-4745SergeevStepan V.СергеевС. В.

PhD student of Department of Computational Mathematics and Artificial Intelligence

1142220124@rudn.ruRUDN UniversityРоссийский университет дружбы народовJoint Institute for Nuclear ResearchОбъединённый институт ядерных исследований15122024324VOL 32, NO4 (2024)ТОМ 32, №4 (2024)41442405042025Copyright ©; 2024, Lovetskiy K.P., Malykh M.D., Sevastianov L.A., Sergeev S.V.Copyright ©; 2024, Ловецкий К.П., Малых М.Д., Севастьянов Л.А., Сергеев С.В.2024Lovetskiy K.P., Malykh M.D., Sevastianov L.A., Sergeev S.V.Ловецкий К.П., Малых М.Д., Севастьянов Л.А., Сергеев С.В.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/43670

Earlier we developed a stable fast numerical algorithm for solving ordinary differential equations of the first order. The method based on the Chebyshev collocation allows solving both initial value problems and problems with a fixed condition at an arbitrary point of the interval with equal success. The algorithm for solving the boundary value problem practically implements a single-pass analogue of the shooting method traditionally used in such cases. In this paper, we extend the developed algorithm to the class of linear ODEs of the second order. Active use of the method of integrating factors and the d’Alembert method allows us to reduce the method for solving second-order equations to a sequence of solutions of a pair of first-order equations. The general solution of the initial or boundary value problem for an inhomogeneous equation of the second order is represented as a sum of basic solutions with unknown constant coefficients. This approach ensures numerical stability, clarity, and simplicity of the algorithm.

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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