Solving a two-point second-order LODE problem by constructing a complete system of solutions using a modified Chebyshev collocation method
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...
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| Format: | Article |
| Language: | English |
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Peoples’ Friendship University of Russia (RUDN University)
2024-12-01
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| Series: | Discrete and Continuous Models and Applied Computational Science |
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| Online Access: | https://journals.rudn.ru/miph/article/viewFile/43670/24667 |
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| author | Konstantin P. Lovetskiy Mikhail D. Malykh Leonid A. Sevastianov Stepan V. Sergeev |
| author_facet | Konstantin P. Lovetskiy Mikhail D. Malykh Leonid A. Sevastianov Stepan V. Sergeev |
| author_sort | Konstantin P. Lovetskiy |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-df6355247bc64c3c998c254b1b08c5d1 |
| institution | OA Journals |
| issn | 2658-4670 2658-7149 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Peoples’ Friendship University of Russia (RUDN University) |
| record_format | Article |
| series | Discrete and Continuous Models and Applied Computational Science |
| spelling | doaj-art-df6355247bc64c3c998c254b1b08c5d12025-08-20T02:26:02ZengPeoples’ Friendship University of Russia (RUDN University)Discrete and Continuous Models and Applied Computational Science2658-46702658-71492024-12-0132441442410.22363/2658-4670-2024-32-4-414-42421067Solving a two-point second-order LODE problem by constructing a complete system of solutions using a modified Chebyshev collocation methodKonstantin P. Lovetskiy0https://orcid.org/0000-0002-3645-1060Mikhail D. Malykh1https://orcid.org/0000-0001-6541-6603Leonid A. Sevastianov2https://orcid.org/0000-0002-1856-4643Stepan V. Sergeev3https://orcid.org/0009-0004-1159-4745RUDN UniversityRUDN UniversityRUDN UniversityRUDN UniversityEarlier 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.https://journals.rudn.ru/miph/article/viewFile/43670/24667linear ordinary differential equation of the second orderstable methodchebyshev collocation methodd’alembert methodintegrating factor |
| spellingShingle | Konstantin P. Lovetskiy Mikhail D. Malykh Leonid A. Sevastianov Stepan V. Sergeev Solving a two-point second-order LODE problem by constructing a complete system of solutions using a modified Chebyshev collocation method Discrete and Continuous Models and Applied Computational Science linear ordinary differential equation of the second order stable method chebyshev collocation method d’alembert method integrating factor |
| title | Solving a two-point second-order LODE problem by constructing a complete system of solutions using a modified Chebyshev collocation method |
| title_full | Solving a two-point second-order LODE problem by constructing a complete system of solutions using a modified Chebyshev collocation method |
| title_fullStr | Solving a two-point second-order LODE problem by constructing a complete system of solutions using a modified Chebyshev collocation method |
| title_full_unstemmed | Solving a two-point second-order LODE problem by constructing a complete system of solutions using a modified Chebyshev collocation method |
| title_short | Solving a two-point second-order LODE problem by constructing a complete system of solutions using a modified Chebyshev collocation method |
| title_sort | solving a two point second order lode problem by constructing a complete system of solutions using a modified chebyshev collocation method |
| topic | linear ordinary differential equation of the second order stable method chebyshev collocation method d’alembert method integrating factor |
| url | https://journals.rudn.ru/miph/article/viewFile/43670/24667 |
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