On the periodic solutions of the second-order wave equations. V
It is established that the linear problem $u_{tt} - a^2 u_{xx} = g(x, t),\quad u(0, t) = u(\pi, t),\quad u(x, t + T) = u(x, t)$ is always solvable in the space of functions $A = \{g:\; g(x, t) = g(x, t + T) = g(\pi - x, t) = -g(-x, t)\}$ provided that $aTq = (2p - 1)\pi, \quad (2p - 1, q) = 1$, wher...
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| Date: | 1993 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
1993
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/5908 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512128590413824 |
|---|---|
| author | Mitropolskiy, Yu. A. Khoma, G. P. Митропольський, Ю. О. Хома, Г. П. |
| author_facet | Mitropolskiy, Yu. A. Khoma, G. P. Митропольський, Ю. О. Хома, Г. П. |
| author_sort | Mitropolskiy, Yu. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T09:20:26Z |
| description | It is established that the linear problem $u_{tt} - a^2 u_{xx} = g(x, t),\quad u(0, t) = u(\pi, t),\quad u(x, t + T) = u(x, t)$
is always solvable in the space of functions $A = \{g:\; g(x, t) = g(x, t + T) = g(\pi - x, t) = -g(-x, t)\}$
provided that $aTq = (2p - 1)\pi, \quad (2p - 1, q) = 1$, where $p, q$ are integers.
To prove this statement, an explicit solution is constructed in the form of an integral operator which is used to prove the existence of a solution
to aperiodic boundary value problem for nonlinear second order wave equation.
The results obtained can be employed in the study of solutions to nonlinear boundary value problems by asymptotic methods. |
| first_indexed | 2026-03-24T03:23:52Z |
| format | Article |
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| id | umjimathkievua-article-5908 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:23:52Z |
| publishDate | 1993 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b8/210fd12c59bd9a83670c778ae6608fb8.pdf |
| spelling | umjimathkievua-article-59082020-03-19T09:20:26Z On the periodic solutions of the second-order wave equations. V О периодических решениях волновых уравнений второго порядка Mitropolskiy, Yu. A. Khoma, G. P. Митропольський, Ю. О. Хома, Г. П. It is established that the linear problem $u_{tt} - a^2 u_{xx} = g(x, t),\quad u(0, t) = u(\pi, t),\quad u(x, t + T) = u(x, t)$ is always solvable in the space of functions $A = \{g:\; g(x, t) = g(x, t + T) = g(\pi - x, t) = -g(-x, t)\}$ provided that $aTq = (2p - 1)\pi, \quad (2p - 1, q) = 1$, where $p, q$ are integers. To prove this statement, an explicit solution is constructed in the form of an integral operator which is used to prove the existence of a solution to aperiodic boundary value problem for nonlinear second order wave equation. The results obtained can be employed in the study of solutions to nonlinear boundary value problems by asymptotic methods. У просторі функцій $A = \{g:\; g(x, t) = g(x, t + T) = g(\pi - x, t) = -g(-x, t)\}$встановлено, що при виконанні умови $aTq = (2p - 1)\pi, \quad (2p - 1, q) = 1$, де $р, q$ — цілі числа, лінійна задача $u_{tt} - a^2 u_{xx} = g(x, t),\quad u(0, t) = u(\pi, t),\quad u(x, t + T) = u(x, t)$ завжди сумісна. Для доведення цього твердження побудовано точний розв'язок у вигляді інтегрального оператора, який використовується при доведенні існування розв'язку періодичної крайової задачі для нелійного рівняння другого порядку. Одержані результати застосовуються при дослідженні розв'язків нелінійних крайових задач асимптотичними методами. Institute of Mathematics, NAS of Ukraine 1993-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5908 Ukrains’kyi Matematychnyi Zhurnal; Vol. 45 No. 8 (1993); 1115–1121 Український математичний журнал; Том 45 № 8 (1993); 1115–1121 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/5908/8499 https://umj.imath.kiev.ua/index.php/umj/article/view/5908/8500 Copyright (c) 1993 Mitropolskiy Yu. A.; Khoma G. P. |
| spellingShingle | Mitropolskiy, Yu. A. Khoma, G. P. Митропольський, Ю. О. Хома, Г. П. On the periodic solutions of the second-order wave equations. V |
| title | On the periodic solutions of the second-order wave equations. V |
| title_alt | О периодических решениях волновых уравнений второго порядка |
| title_full | On the periodic solutions of the second-order wave equations. V |
| title_fullStr | On the periodic solutions of the second-order wave equations. V |
| title_full_unstemmed | On the periodic solutions of the second-order wave equations. V |
| title_short | On the periodic solutions of the second-order wave equations. V |
| title_sort | on the periodic solutions of the second-order wave equations. v |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5908 |
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