Two-point boundary-value problems for differential equations with generalized piecewise-constant argument
UDC 517.9 We consider  a two-point boundary-value problem for a system of differential equations with generalized piecewise-constant argument. To solve the problem, we propose to use a constructive method based on the Dzhumabaev parametrization method and a new approach to the concept o...
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| Дата: | 2024 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2024
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/7384 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512662752854016 |
|---|---|
| author | Assanova, Anar T. Uteshova, Roza E. Assanova/gmail, Anar Turmaganbetkyzy Assanova, Anar T. Uteshova, Roza E. |
| author_facet | Assanova, Anar T. Uteshova, Roza E. Assanova/gmail, Anar Turmaganbetkyzy Assanova, Anar T. Uteshova, Roza E. |
| author_sort | Assanova, Anar T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-06-27T08:49:56Z |
| description | UDC 517.9
We consider  a two-point boundary-value problem for a system of differential equations with generalized piecewise-constant argument. To solve the problem, we propose to use a constructive method based on the Dzhumabaev parametrization method and a new approach to the concept of general solution. The interval is partitioned with regard for  the singularities of the argument. The values of the solution at the interior points of the partition are regarded as additional parameters, and the differential equation is transformed into a system of Cauchy problems with parameters on subintervals of the partition. By using the solutions of these problems, we obtain a new general solution of the differential equation with piecewise-constant argument and establish its properties. The new general solution, boundary conditions, and the conditions of continuity of the solution at the interior points of the partition are used to construct a linear system of algebraic equations for the introduced parameters. The coefficients and the right-hand side of the system are found as a result of the solution of Cauchy problems for linear ordinary differential equations on the  subintervals of the partition. It is shown that the solvability of the boundary-value problem is equivalent to the solvability of the constructed system. We propose algorithms of the parametrization method for solving the analyzed  boundary-value problem and establish necessary and sufficient conditions for the well-posedness of this problem. |
| doi_str_mv | 10.3842/umzh.v76i5.7384 |
| first_indexed | 2026-03-24T03:32:21Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-7384 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:21Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-73842024-06-27T08:49:56Z Two-point boundary-value problems for differential equations with generalized piecewise-constant argument Two-point boundary-value problems for differential equations with generalized piecewise-constant argument Assanova, Anar T. Uteshova, Roza E. Assanova/gmail, Anar Turmaganbetkyzy Assanova, Anar T. Uteshova, Roza E. Differential equations with generalized piecewise-constant argumen, two-point boundary-value problem, parametrization method, new general solution, solvability criteria. UDC 517.9 We consider  a two-point boundary-value problem for a system of differential equations with generalized piecewise-constant argument. To solve the problem, we propose to use a constructive method based on the Dzhumabaev parametrization method and a new approach to the concept of general solution. The interval is partitioned with regard for  the singularities of the argument. The values of the solution at the interior points of the partition are regarded as additional parameters, and the differential equation is transformed into a system of Cauchy problems with parameters on subintervals of the partition. By using the solutions of these problems, we obtain a new general solution of the differential equation with piecewise-constant argument and establish its properties. The new general solution, boundary conditions, and the conditions of continuity of the solution at the interior points of the partition are used to construct a linear system of algebraic equations for the introduced parameters. The coefficients and the right-hand side of the system are found as a result of the solution of Cauchy problems for linear ordinary differential equations on the  subintervals of the partition. It is shown that the solvability of the boundary-value problem is equivalent to the solvability of the constructed system. We propose algorithms of the parametrization method for solving the analyzed  boundary-value problem and establish necessary and sufficient conditions for the well-posedness of this problem. УДК 517.9 Двоточкові крайові задачі для диференціальних рівнянь з узагальненим кусково-сталим аргументом Розглянуто двоточкову крайову задачу для системи диференціальних рівнянь із узагальненим кусково-сталим аргументом. Для її розв'язання запропоновано конструктивний метод, що базується на методі параметризації Джумабаєва та новому підході до поняття загального розв'язку. Інтервал розбито з урахуванням особливостей аргументу. Значення розв'язку у внутрішніх точках розбиття розглянуто як додаткові параметри, а диференціальне рівняння перетворено на систему задач Коші з параметрами, заданими на підінтервалах розбиття. За допомогою розв'язків цих задач отримано новий загальний розв'язок диференціального рівняння з кусково-сталим аргументом і встановлено його властивості. Новий загальний розв'язок, граничні умови та умови неперервності розв'язку у внутрішніх точках розбиття використовують для побудови лінійної системи алгебраїчних рівнянь щодо введених параметрів. Коефіцієнти та праву частину системи знайдено шляхом розв'язування задач Коші для лінійних звичайних диференціальних рівнянь на підінтервалах розбиття. Показано, що розв'язність крайової задачі еквівалентна розв'язності побудованої системи. Запропоновано алгоритми методу параметризації для розв'язування досліджуваної крайової задачі та встановлено необхідні й достатні умови її коректної постановки. Institute of Mathematics, NAS of Ukraine 2024-06-02 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7384 10.3842/umzh.v76i5.7384 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 5 (2024); 631 - 646 Український математичний журнал; Том 76 № 5 (2024); 631 - 646 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7384/9936 Copyright (c) 2024 Anar Turmaganbetkyzy Assanova/gmail |
| spellingShingle | Assanova, Anar T. Uteshova, Roza E. Assanova/gmail, Anar Turmaganbetkyzy Assanova, Anar T. Uteshova, Roza E. Two-point boundary-value problems for differential equations with generalized piecewise-constant argument |
| title | Two-point boundary-value problems for differential equations with generalized piecewise-constant argument |
| title_alt | Two-point boundary-value problems for differential equations with generalized piecewise-constant argument |
| title_full | Two-point boundary-value problems for differential equations with generalized piecewise-constant argument |
| title_fullStr | Two-point boundary-value problems for differential equations with generalized piecewise-constant argument |
| title_full_unstemmed | Two-point boundary-value problems for differential equations with generalized piecewise-constant argument |
| title_short | Two-point boundary-value problems for differential equations with generalized piecewise-constant argument |
| title_sort | two-point boundary-value problems for differential equations with generalized piecewise-constant argument |
| topic_facet | Differential equations with generalized piecewise-constant argumen two-point boundary-value problem parametrization method new general solution solvability criteria. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7384 |
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