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)3511110.22363/2658-4670-2023-31-2-150-163WFZCIOArticlesСтатьиResearch ArticleChebyshev collocation method for solving second order ODEs using integration matricesМетод коллокации Чебышева для решения ОДУ второго порядка с использованием матриц интегрированияhttps://orcid.org/0000-0002-3645-1060LovetskiyKonstantin P.ЛовецкийК. П.

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Applied Probability and Informatics

lovetskiy-kp@rudn.ruhttps://orcid.org/0000-0002-0877-7063KulyabovDmitry S.КулябовД. С.

Professor, Doctor of Sciences in Physics and Mathematics, Professor at the Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University); Senior Researcher of Laboratory of Information Technologies, Joint Institute for Nuclear Research

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

Professor, Doctor of Sciences in Physics and Mathematics, Professor at the Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia named after Patrice Lumumba (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 Applied Probability and Informatics

1142220124@rudn.ruRUDN UniversityРоссийский университет дружбы народовJoint Institute for Nuclear ResearchОбъединённый институт ядерных исследований30062023312VOL 31, NO2 (2023)ТОМ 31, №2 (2023)15016329062023Copyright ©; 2023, Lovetskiy K.P., Kulyabov D.S., Sevastianov L.A., Sergeev S.V.Copyright ©; 2023, Ловецкий К.П., Кулябов Д.С., Севастьянов Л.А., Сергеев С.В.2023Lovetskiy K.P., Kulyabov D.S., Sevastianov L.A., Sergeev S.V.Ловецкий К.П., Кулябов Д.С., Севастьянов Л.А., Сергеев С.В.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/35111

The spectral collocation method for solving two-point boundary value problems for second order differential equations is implemented, based on representing the solution as an expansion in Chebyshev polynomials. The approach allows a stable calculation of both the spectral representation of the solution and its pointwise representation on any required grid in the definition domain of the equation and additional conditions of the multipoint problem. For the effective construction of SLAE, the solution of which gives the desired coefficients, the Chebyshev matrices of spectral integration are actively used. The proposed algorithms have a high accuracy for moderate-dimension systems of linear algebraic equations. The matrix of the system remains well-conditioned and, with an increase in the number of collocation points, allows finding solutions with ever-increasing accuracy.

Реализован метод спектральной коллокации для решения двухточечных краевых задач для дифференциальных уравнений второго порядка, основанный на представлении решения в виде разложения по полиномам Чебышева. Подход позволяет устойчиво вычислять как спектральное представление решения, так и его поточечное представление на любой необходимой сетке в области определения уравнения и дополнительных условий многоточечной задачи. Для эффективного построения СЛАУ, решение которой дает искомые коэффициенты, активно используются матрицы Чебышева спектрального интегрирования. Предложенные алгоритмы обладают высокой точностью для систем линейных алгебраических уравнений средней размерности. Матрица системы остается хорошо обусловленной и с увеличением количества точек коллокации позволяет находить решения со все возрастающей точностью.

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