On two four-dimensional curl operators and their applications
Academician O.A. Ladyzhenskaya emphasized the importance of constructing a fundamental system in the space of solenoidal functions for simple domains such as squares, cubes, and similar regions. This article examines the problem of constructing such fundamental systems for a four-dimensional parall...
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| Format: | Article |
| Language: | English |
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Academician Ye.A. Buketov Karaganda University
2025-06-01
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| Series: | Қарағанды университетінің хабаршысы. Математика сериясы |
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| Online Access: | https://mts.buketov.edu.kz/index.php/mathematics-vestnik/article/view/909 |
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| _version_ | 1849471352677859328 |
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| author | M.T. Jenaliyev A.S. Kassymbekova M.G. Yergaliyev |
| author_facet | M.T. Jenaliyev A.S. Kassymbekova M.G. Yergaliyev |
| author_sort | M.T. Jenaliyev |
| collection | DOAJ |
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Academician O.A. Ladyzhenskaya emphasized the importance of constructing a fundamental system in the space of solenoidal functions for simple domains such as squares, cubes, and similar regions. This article examines the problem of constructing such fundamental systems for a four-dimensional parallelepiped and cube. As is well known, applying the stream functions known from the two- and three-dimensional cases, the spectral problem for the Stokes operator reduces to the so-called clamped plate problem, which, in turn, has no solution in domains such as the square, cube, or parallelepiped. Thus, in higher-dimensional cases, the necessity of an analogous stream function becomes evident. In this work, the authors propose two curl operators that satisfy the above-mentioned requirements. Using the introduced curl operators, the spectral problem for the biharmonic operator in a four-dimensional parallelepiped and cube is formulated. Alternative approaches to constructing a fundamental system are presented, given the unsolvability of the spectral problem. Furthermore, the growth orders of the obtained eigenvalues are established.
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| format | Article |
| id | doaj-art-440f5b551b514775a8cde494fa8582bd |
| institution | Kabale University |
| issn | 2518-7929 2663-5011 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | Academician Ye.A. Buketov Karaganda University |
| record_format | Article |
| series | Қарағанды университетінің хабаршысы. Математика сериясы |
| spelling | doaj-art-440f5b551b514775a8cde494fa8582bd2025-08-20T03:24:52ZengAcademician Ye.A. Buketov Karaganda UniversityҚарағанды университетінің хабаршысы. Математика сериясы2518-79292663-50112025-06-01118210.31489/2025m2/106-121On two four-dimensional curl operators and their applicationsM.T. Jenaliyev0https://orcid.org/0000-0001-8743-7026A.S. Kassymbekova1https://orcid.org/0000-0002-4105-625XM.G. Yergaliyev2https://orcid.org/0000-0001-8638-4647Institute of Mathematics and Mathematical Modeling, Almaty, KazakhstanInstitute of Mathematics and Mathematical Modeling, Almaty, KazakhstanInstitute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan Academician O.A. Ladyzhenskaya emphasized the importance of constructing a fundamental system in the space of solenoidal functions for simple domains such as squares, cubes, and similar regions. This article examines the problem of constructing such fundamental systems for a four-dimensional parallelepiped and cube. As is well known, applying the stream functions known from the two- and three-dimensional cases, the spectral problem for the Stokes operator reduces to the so-called clamped plate problem, which, in turn, has no solution in domains such as the square, cube, or parallelepiped. Thus, in higher-dimensional cases, the necessity of an analogous stream function becomes evident. In this work, the authors propose two curl operators that satisfy the above-mentioned requirements. Using the introduced curl operators, the spectral problem for the biharmonic operator in a four-dimensional parallelepiped and cube is formulated. Alternative approaches to constructing a fundamental system are presented, given the unsolvability of the spectral problem. Furthermore, the growth orders of the obtained eigenvalues are established. https://mts.buketov.edu.kz/index.php/mathematics-vestnik/article/view/909spectral problemfundamental systemcurl operator |
| spellingShingle | M.T. Jenaliyev A.S. Kassymbekova M.G. Yergaliyev On two four-dimensional curl operators and their applications Қарағанды университетінің хабаршысы. Математика сериясы spectral problem fundamental system curl operator |
| title | On two four-dimensional curl operators and their applications |
| title_full | On two four-dimensional curl operators and their applications |
| title_fullStr | On two four-dimensional curl operators and their applications |
| title_full_unstemmed | On two four-dimensional curl operators and their applications |
| title_short | On two four-dimensional curl operators and their applications |
| title_sort | on two four dimensional curl operators and their applications |
| topic | spectral problem fundamental system curl operator |
| url | https://mts.buketov.edu.kz/index.php/mathematics-vestnik/article/view/909 |
| work_keys_str_mv | AT mtjenaliyev ontwofourdimensionalcurloperatorsandtheirapplications AT askassymbekova ontwofourdimensionalcurloperatorsandtheirapplications AT mgyergaliyev ontwofourdimensionalcurloperatorsandtheirapplications |