Investigation of the approximate solution of one class of curvilinear integral equations by the projection method
UDC 517.9 We prove the existence theorem for the normal derivative of the double-layer potential and establish the formula for its evaluation. A new method for the construction of quadrature formulas for the normal derivatives of simple- and double-layer potentials is developed, and the...
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| Дата: | 2025 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2025
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/7762 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512737019297792 |
|---|---|
| author | Khalilov, Elnur H. Aliev, Araz R. Musayev, Ali M. Khalilov, Elnur H. Aliev, Araz R. Musayev, Ali M. |
| author_facet | Khalilov, Elnur H. Aliev, Araz R. Musayev, Ali M. Khalilov, Elnur H. Aliev, Araz R. Musayev, Ali M. |
| author_sort | Khalilov, Elnur H. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-05-09T12:53:49Z |
| description | UDC 517.9
We prove the existence theorem for the normal derivative of the double-layer potential and establish the formula for its evaluation. A new method for the construction of quadrature formulas for the normal derivatives of simple- and double-layer potentials is developed, and the error estimates are obtained for the constructed quadrature formulas. By using these quadrature formulas, the integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation in two-dimensional space is replaced by a system of algebraic equations, and the existence and uniqueness of the solution to this system is proved.  The convergence of the solution of the system of algebraic equations to the exact solution of the integral equation at the control points is proved and the convergence rate of the method is determined. |
| doi_str_mv | 10.3842/umzh.v76i10.7762 |
| first_indexed | 2026-03-24T03:33:32Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-7762 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:33:32Z |
| publishDate | 2025 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-77622025-05-09T12:53:49Z Investigation of the approximate solution of one class of curvilinear integral equations by the projection method Investigation of the approximate solution of one class of curvilinear integral equations by the projection method Khalilov, Elnur H. Aliev, Araz R. Musayev, Ali M. Khalilov, Elnur H. Aliev, Araz R. Musayev, Ali M. exterior Dirichlet boundary value problem Helmholtz equation boundary integral equations method curvilinear singular integral collocation method UDC 517.9 We prove the existence theorem for the normal derivative of the double-layer potential and establish the formula for its evaluation. A new method for the construction of quadrature formulas for the normal derivatives of simple- and double-layer potentials is developed, and the error estimates are obtained for the constructed quadrature formulas. By using these quadrature formulas, the integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation in two-dimensional space is replaced by a system of algebraic equations, and the existence and uniqueness of the solution to this system is proved.  The convergence of the solution of the system of algebraic equations to the exact solution of the integral equation at the control points is proved and the convergence rate of the method is determined. УДК 517.9 Дослідження наближеного розв’язку одного класу криволінійних інтегральних рівнянь проєкційним методом Доведено теорему існування нормальної похідної потенціалу подвійного шару та наведено формулу для її обчислення. Розроблено новий метод побудови квадратурних формул для нормальних похідних від потенціалів простого та подвійного шарів та одержано оцінки для похибок отриманих квадратурних формул. На основі цих квадратурних формул інтегральне рівняння зовнішньої крайової задачі Діріхле для рівняння Гельмгольца у двовимірному просторі замінено на систему алгебраїчних рівнянь. Доведено існування та єдиність розв’язку цієї системи. Крім того, доведено збіжність розв’язку системи алгебраїчних рівнянь до точного розв’язку інтегрального рівняння в  контрольних точках та знайдено швидкість цієї збіжності. Institute of Mathematics, NAS of Ukraine 2025-05-07 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7762 10.3842/umzh.v76i10.7762 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 10 (2024); 1543 - 1564 Український математичний журнал; Том 76 № 10 (2024); 1543 - 1564 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7762/10222 Copyright (c) 2024 Elnur Khalilov |
| spellingShingle | Khalilov, Elnur H. Aliev, Araz R. Musayev, Ali M. Khalilov, Elnur H. Aliev, Araz R. Musayev, Ali M. Investigation of the approximate solution of one class of curvilinear integral equations by the projection method |
| title | Investigation of the approximate solution of one class of curvilinear integral equations by the projection method |
| title_alt | Investigation of the approximate solution of one class of curvilinear integral equations by the projection method |
| title_full | Investigation of the approximate solution of one class of curvilinear integral equations by the projection method |
| title_fullStr | Investigation of the approximate solution of one class of curvilinear integral equations by the projection method |
| title_full_unstemmed | Investigation of the approximate solution of one class of curvilinear integral equations by the projection method |
| title_short | Investigation of the approximate solution of one class of curvilinear integral equations by the projection method |
| title_sort | investigation of the approximate solution of one class of curvilinear integral equations by the projection method |
| topic_facet | exterior Dirichlet boundary value problem Helmholtz equation boundary integral equations method curvilinear singular integral collocation method |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7762 |
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