Legendre superconvergent degenerate kernel and Nyström methods for nonlinear integral equations
UDC 517.9 We study polynomially based superconvergent collocation methods for the approximation of solutions of nonlinear integral equations. The superconvergent degenerate kernel method is chosen for approximating the solutions of Hammerstein equations, while a superconvergent Ny...
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Institute of Mathematics, NAS of Ukraine
2023
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512580719607808 |
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| author | Allouch, C. Arrai, M. Bouda, H. Tahrichi, M. Allouch, C. Arrai, M. Bouda, H. Tahrichi, M. |
| author_facet | Allouch, C. Arrai, M. Bouda, H. Tahrichi, M. Allouch, C. Arrai, M. Bouda, H. Tahrichi, M. |
| author_sort | Allouch, C. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-06-13T15:56:16Z |
| description |
UDC 517.9
We study polynomially based superconvergent collocation methods for the approximation of solutions of nonlinear integral equations. The superconvergent degenerate kernel method is chosen for approximating the solutions of Hammerstein equations, while a superconvergent Nystr\"om method is used for solving Urysohn equations. By applying  interpolatory projections based on Legendre polynomials of degree $\leq n,$ we analyze the superconvergence of these methods and their iterated versions.  Numerical results are presented to validate the theoretical results. |
| doi_str_mv | 10.37863/umzh.v75i5.7039 |
| first_indexed | 2026-03-24T03:31:03Z |
| format | Article |
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| id | umjimathkievua-article-7039 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:31:03Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-70392023-06-13T15:56:16Z Legendre superconvergent degenerate kernel and Nyström methods for nonlinear integral equations Legendre superconvergent degenerate kernel and Nyström methods for nonlinear integral equations Allouch, C. Arrai, M. Bouda, H. Tahrichi, M. Allouch, C. Arrai, M. Bouda, H. Tahrichi, M. Hammerstein equation, Urysohn equation, Degenerate kernel method, Nystr¨om method, Interpolatory projection, Legendre polynomial, Superconvergence. UDC 517.9 We study polynomially based superconvergent collocation methods for the approximation of solutions of nonlinear integral equations. The superconvergent degenerate kernel method is chosen for approximating the solutions of Hammerstein equations, while a superconvergent Nystr\"om method is used for solving Urysohn equations. By applying  interpolatory projections based on Legendre polynomials of degree $\leq n,$ we analyze the superconvergence of these methods and their iterated versions.  Numerical results are presented to validate the theoretical results. УДК 517.9 Суперзбіжне вироджене ядро Лежандра i методи Ністрема для нелінійних інтегральних рівнянь  Досліджено суперзбіжні методи колокації на поліноміальній основі для апроксимації розв’язків нелінійних інтегральних рівнянь. Для апроксимації розв’язків рівнянь Гаммерштейна використано суперзбіжний метод виродженого ядра, а для розв’язування рівнянь Урисона — суперзбіжний метод  Ністрема. Застосовуючи інтерполяційні проєкції на основі поліномів Лежандра   степеня $\leq n,$ проаналізовано суперзбіжність цих методів та їхніх ітерованих версій. Дані числових розрахунків наведено для підтвердження теоретичних результатів.  Institute of Mathematics, NAS of Ukraine 2023-05-24 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7039 10.37863/umzh.v75i5.7039 Ukrains’kyi Matematychnyi Zhurnal; Vol. 75 No. 5 (2023); 579 - 595 Український математичний журнал; Том 75 № 5 (2023); 579 - 595 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7039/9769 Copyright (c) 2023 Chafik |
| spellingShingle | Allouch, C. Arrai, M. Bouda, H. Tahrichi, M. Allouch, C. Arrai, M. Bouda, H. Tahrichi, M. Legendre superconvergent degenerate kernel and Nyström methods for nonlinear integral equations |
| title | Legendre superconvergent degenerate kernel and Nyström methods for nonlinear integral equations |
| title_alt | Legendre superconvergent degenerate kernel and Nyström methods for nonlinear integral equations |
| title_full | Legendre superconvergent degenerate kernel and Nyström methods for nonlinear integral equations |
| title_fullStr | Legendre superconvergent degenerate kernel and Nyström methods for nonlinear integral equations |
| title_full_unstemmed | Legendre superconvergent degenerate kernel and Nyström methods for nonlinear integral equations |
| title_short | Legendre superconvergent degenerate kernel and Nyström methods for nonlinear integral equations |
| title_sort | legendre superconvergent degenerate kernel and nyström methods for nonlinear integral equations |
| topic_facet | Hammerstein equation Urysohn equation Degenerate kernel method Nystr¨om method Interpolatory projection Legendre polynomial Superconvergence. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7039 |
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