Asymptotic Integration of Certain Differential Equations in Banach Space
We investigate the problem of constructing the asymptotics for weak solutions of certain class of linear differential equations in the Banach space as the independent variable tends to infinity. The studied class of equations is the perturbation of linear autonomous equation, generally speaking, wit...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2017-10-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/583 |
| _version_ | 1797878007851909120 |
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| author | Pavel N. Nesterov |
| author_facet | Pavel N. Nesterov |
| author_sort | Pavel N. Nesterov |
| collection | DOAJ |
| description | We investigate the problem of constructing the asymptotics for weak solutions of certain class of linear differential equations in the Banach space as the independent variable tends to infinity. The studied class of equations is the perturbation of linear autonomous equation, generally speaking, with an unbounded operator. The perturbation takes the form of the family of the bounded operators that, in a sense, decreases oscillatory at infinity. The unperturbed equation satisfies the standard requirements of the center manifold theory. The essence of the proposed asymptotic integration method is to prove the existence for the initial equation of the center-like manifold (critical manifold). This manifold is positively invariant with respect to the initial equation and attracts all the trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional ordinary differential system. The asymptotics for the fundamental matrix of this system may be constructed by using the method proposed by the author for asymptotic integration of the systems with oscillatory decreasing coefficients. We illustrate the suggested technique by constructing the asymptotic formulas for solutions of the perturbed heat equation. |
| first_indexed | 2024-04-10T02:25:58Z |
| format | Article |
| id | doaj.art-4a1678b347e249c98cb01d500575429f |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-10T02:25:58Z |
| publishDate | 2017-10-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-4a1678b347e249c98cb01d500575429f2023-03-13T08:07:29ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172017-10-0124559661410.18255/1818-1015-2017-5-596-614425Asymptotic Integration of Certain Differential Equations in Banach SpacePavel N. Nesterov0Ярославский государственный университет им. П.Г. ДемидоваWe investigate the problem of constructing the asymptotics for weak solutions of certain class of linear differential equations in the Banach space as the independent variable tends to infinity. The studied class of equations is the perturbation of linear autonomous equation, generally speaking, with an unbounded operator. The perturbation takes the form of the family of the bounded operators that, in a sense, decreases oscillatory at infinity. The unperturbed equation satisfies the standard requirements of the center manifold theory. The essence of the proposed asymptotic integration method is to prove the existence for the initial equation of the center-like manifold (critical manifold). This manifold is positively invariant with respect to the initial equation and attracts all the trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional ordinary differential system. The asymptotics for the fundamental matrix of this system may be constructed by using the method proposed by the author for asymptotic integration of the systems with oscillatory decreasing coefficients. We illustrate the suggested technique by constructing the asymptotic formulas for solutions of the perturbed heat equation.https://www.mais-journal.ru/jour/article/view/583асимптотикадифференциальное уравнениебанахово пространствоколебательно убывающие коэффициентыметод центральных многообразийвозмущенное уравнение теплопроводности |
| spellingShingle | Pavel N. Nesterov Asymptotic Integration of Certain Differential Equations in Banach Space Моделирование и анализ информационных систем асимптотика дифференциальное уравнение банахово пространство колебательно убывающие коэффициенты метод центральных многообразий возмущенное уравнение теплопроводности |
| title | Asymptotic Integration of Certain Differential Equations in Banach Space |
| title_full | Asymptotic Integration of Certain Differential Equations in Banach Space |
| title_fullStr | Asymptotic Integration of Certain Differential Equations in Banach Space |
| title_full_unstemmed | Asymptotic Integration of Certain Differential Equations in Banach Space |
| title_short | Asymptotic Integration of Certain Differential Equations in Banach Space |
| title_sort | asymptotic integration of certain differential equations in banach space |
| topic | асимптотика дифференциальное уравнение банахово пространство колебательно убывающие коэффициенты метод центральных многообразий возмущенное уравнение теплопроводности |
| url | https://www.mais-journal.ru/jour/article/view/583 |
| work_keys_str_mv | AT pavelnnesterov asymptoticintegrationofcertaindifferentialequationsinbanachspace |