On the approximation of solutions of one singular differential equation on the axis
In this paper we study the problem of the best approximation by linear methods of solutions to one Triebel-type equation. This problem was solved by using estimates of the linear widths of the unit ball in corresponding spaces of differentiable functions. According to the definition, linear widths...
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| Format: | Article |
| Language: | English |
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Academician Ye.A. Buketov Karaganda University
2022-12-01
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| Series: | Қарағанды университетінің хабаршысы. Математика сериясы |
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| Online Access: | http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/536 |
| _version_ | 1827395702692511744 |
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| author | A. S. Kassym L. K. Kussainova |
| author_facet | A. S. Kassym L. K. Kussainova |
| author_sort | A. S. Kassym |
| collection | DOAJ |
| description |
In this paper we study the problem of the best approximation by linear methods of solutions to one Triebel-type equation. This problem was solved by using estimates of the linear widths of the unit ball in corresponding spaces of differentiable functions. According to the definition, linear widths give the best
estimates for the approximation of compact sets in a given normed space by linear methods which are implemented through finite-dimensional operators. The problem includes answers to the questions about the solvability of the studied equation, the construction of the corresponding weighted space of differentiable functions, the development of a method for estimating linear widths of compact sets in weighted polynomial Sobolev space. In this work, conditions are obtained under which the considered operator has a bounded
inverse. The weighted Sobolev space corresponding to the posed problem is determined. Upper estimates are obtained for the counting function for a sequence of linear widths, which correspond to the posed problem.
One example is constructed in which two-sided estimates of linear widths are given. The method for solving this problem can be applied to the numerical solution of non-standard ordinary differential equations on an infinite axis.
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| first_indexed | 2024-03-08T18:38:34Z |
| format | Article |
| id | doaj.art-0472fbe452644d81ad7acb082e0fd350 |
| institution | Directory Open Access Journal |
| issn | 2518-7929 2663-5011 |
| language | English |
| last_indexed | 2024-03-08T18:38:34Z |
| publishDate | 2022-12-01 |
| publisher | Academician Ye.A. Buketov Karaganda University |
| record_format | Article |
| series | Қарағанды университетінің хабаршысы. Математика сериясы |
| spelling | doaj.art-0472fbe452644d81ad7acb082e0fd3502023-12-29T10:19:09ZengAcademician Ye.A. Buketov Karaganda UniversityҚарағанды университетінің хабаршысы. Математика сериясы2518-79292663-50112022-12-01108410.31489/2022m/86-97On the approximation of solutions of one singular differential equation on the axisA. S. KassymL. K. Kussainova In this paper we study the problem of the best approximation by linear methods of solutions to one Triebel-type equation. This problem was solved by using estimates of the linear widths of the unit ball in corresponding spaces of differentiable functions. According to the definition, linear widths give the best estimates for the approximation of compact sets in a given normed space by linear methods which are implemented through finite-dimensional operators. The problem includes answers to the questions about the solvability of the studied equation, the construction of the corresponding weighted space of differentiable functions, the development of a method for estimating linear widths of compact sets in weighted polynomial Sobolev space. In this work, conditions are obtained under which the considered operator has a bounded inverse. The weighted Sobolev space corresponding to the posed problem is determined. Upper estimates are obtained for the counting function for a sequence of linear widths, which correspond to the posed problem. One example is constructed in which two-sided estimates of linear widths are given. The method for solving this problem can be applied to the numerical solution of non-standard ordinary differential equations on an infinite axis. http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/536differential equationsTriebel equationsapproximation of sets by linear methodswidths of setsweighted Sobolev spaces |
| spellingShingle | A. S. Kassym L. K. Kussainova On the approximation of solutions of one singular differential equation on the axis Қарағанды университетінің хабаршысы. Математика сериясы differential equations Triebel equations approximation of sets by linear methods widths of sets weighted Sobolev spaces |
| title | On the approximation of solutions of one singular differential equation on the axis |
| title_full | On the approximation of solutions of one singular differential equation on the axis |
| title_fullStr | On the approximation of solutions of one singular differential equation on the axis |
| title_full_unstemmed | On the approximation of solutions of one singular differential equation on the axis |
| title_short | On the approximation of solutions of one singular differential equation on the axis |
| title_sort | on the approximation of solutions of one singular differential equation on the axis |
| topic | differential equations Triebel equations approximation of sets by linear methods widths of sets weighted Sobolev spaces |
| url | http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/536 |
| work_keys_str_mv | AT askassym ontheapproximationofsolutionsofonesingulardifferentialequationontheaxis AT lkkussainova ontheapproximationofsolutionsofonesingulardifferentialequationontheaxis |