Construction of asymptotic solutions of linear singularly disturbed second-order system with degenerations
The paper discusses the asymptotic behavior of the general solution of a linear singularly perturbed system $\varepsilon ^{2h} A(t,\varepsilon )\frac{{d^2 x}}{{dt^2 }} + \varepsilon ^h B(t,\varepsilon )\frac{{dx}}{{dt^2 }} + C(t,\varepsilon )x = 0,$  where x∈Rn, t∈[t0; T], h∈N, ε∈ (0, ε...
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| Date: | 1992 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian |
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Institute of Mathematics, NAS of Ukraine
1992
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/8040 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512884756316160 |
|---|---|
| author | Yakovets , V. P. Яковец , В. П. |
| author_facet | Yakovets , V. P. Яковец , В. П. |
| author_sort | Yakovets , V. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-12-28T09:25:14Z |
| description | The paper discusses the asymptotic behavior of the general solution of a linear singularly perturbed system
$\varepsilon ^{2h} A(t,\varepsilon )\frac{{d^2 x}}{{dt^2 }} + \varepsilon ^h B(t,\varepsilon )\frac{{dx}}{{dt^2 }} + C(t,\varepsilon )x = 0,$
 where x∈Rn, t∈[t0; T], h∈N, ε∈ (0, ε0], ε≪1 and det A(t, 0) ≡ 0. It is assumed that the quadratic pencil of matrices C(t, 0) + λB(t, 0) + λ2A(t, 0) is regular and has either simple “finite” and “infinite” elementary divisors or just one multiple elementary divisor. |
| first_indexed | 2026-03-24T03:35:53Z |
| format | Article |
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| id | umjimathkievua-article-8040 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:35:53Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4f/d285fb10b9953419ba11e153a5d70b4f.pdf |
| spelling | umjimathkievua-article-80402023-12-28T09:25:14Z Construction of asymptotic solutions of linear singularly disturbed second-order system with degenerations Построение асимптотических решений линейных сингулярно возмущенных систем второго порядка с вырождением Yakovets , V. P. Яковец , В. П. The paper discusses the asymptotic behavior of the general solution of a linear singularly perturbed system $\varepsilon ^{2h} A(t,\varepsilon )\frac{{d^2 x}}{{dt^2 }} + \varepsilon ^h B(t,\varepsilon )\frac{{dx}}{{dt^2 }} + C(t,\varepsilon )x = 0,$  where x∈Rn, t∈[t0; T], h∈N, ε∈ (0, ε0], ε≪1 and det A(t, 0) ≡ 0. It is assumed that the quadratic pencil of matrices C(t, 0) + λB(t, 0) + λ2A(t, 0) is regular and has either simple “finite” and “infinite” elementary divisors or just one multiple elementary divisor. Исследован вопрос о построении асимптотики общего решения линейной сингулярно возмущенной системы вида $\varepsilon ^{2h} A(t,\varepsilon )\frac{{d^2 x}}{{dt^2 }} + \varepsilon ^h B(t,\varepsilon )\frac{{dx}}{{dt^2 }} + C(t,\varepsilon )x = 0,$ где x∈Rn, t∈[t0; T], h∈N, ε∈ (0, ε0], ε≪1 и det A(t, 0) ≡ 0. Рассмотрен случай, когда квадратичный пучок матриц C(t, 0) + λB(t, 0) + λ2A(t, 0) регулярный и имеет либо простые «конечные» и «бесконечный» элементарные делители, либо по одному кратному. Досліджене питання про побудову асимптотики загального розв’язку лінійної сингулярно збуреної системи вигляду $\varepsilon ^{2h} A(t,\varepsilon )\frac{{d^2 x}}{{dt^2 }} + \varepsilon ^h B(t,\varepsilon )\frac{{dx}}{{dt^2 }} + C(t,\varepsilon )x = 0,$ де x∈Rn, t∈[t0; T], h∈N, ε∈ (0, ε0], ε≪1 і det A(t, 0) ≡ 0. Розглянуто випадок, коли квадратичний пучок матриць C(t, 0) + λB(t, 0) + λ2A(t, 0) регулярний і має або прості «скінченні» і «нескінченний» елементарні дільники, або по одному кратному. Institute of Mathematics, NAS of Ukraine 1992-09-08 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/8040 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 8 (1992); 1102-1112 Український математичний журнал; Том 44 № 8 (1992); 1102-1112 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/8040/9603 Copyright (c) 1992 V. P. Yakovets |
| spellingShingle | Yakovets , V. P. Яковец , В. П. Construction of asymptotic solutions of linear singularly disturbed second-order system with degenerations |
| title | Construction of asymptotic solutions of linear singularly disturbed second-order system with degenerations |
| title_alt | Построение асимптотических решений линейных сингулярно возмущенных систем второго порядка с вырождением |
| title_full | Construction of asymptotic solutions of linear singularly disturbed second-order system with degenerations |
| title_fullStr | Construction of asymptotic solutions of linear singularly disturbed second-order system with degenerations |
| title_full_unstemmed | Construction of asymptotic solutions of linear singularly disturbed second-order system with degenerations |
| title_short | Construction of asymptotic solutions of linear singularly disturbed second-order system with degenerations |
| title_sort | construction of asymptotic solutions of linear singularly disturbed second-order system with degenerations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8040 |
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