Composite trapezoidal quadrature for computing hypersingular integrals on interval
In this paper, composite trapezoidal quadrature for numerical evaluation of hypersingular integrals was first introduced. By Taylor expansion at the singular point $ y $, error functional was obtained. We know that the divergence rate of $ O(h^{-p}), p = 1, 2 $, and there were no roots of the specia...
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AIMS Press
2024-12-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241645 |
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| author | Xiaoping Zhang Jin Li |
| author_facet | Xiaoping Zhang Jin Li |
| author_sort | Xiaoping Zhang |
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| description | In this paper, composite trapezoidal quadrature for numerical evaluation of hypersingular integrals was first introduced. By Taylor expansion at the singular point $ y $, error functional was obtained. We know that the divergence rate of $ O(h^{-p}), p = 1, 2 $, and there were no roots of the special function for the first part in the error functional. Meanwhile, for the second part of the error functional, the divergence rate was $ O(h^{-p+1}), p = 1, 2 $, but there were roots of the special function. We proved that the convergence rate could reach $ O(h^{2}) $ at superconvergence points far from the end of the interval. Two modified trapezoidal quadratures are presented and their convergence rate can reach $ O(h^{2}) $ at certain superconvergence points or any local coordinate point. At last, several examples were presented to test our theorem. |
| format | Article |
| id | doaj-art-6e8769f1cf9a464cb43d2e0f845c5dea |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-6e8769f1cf9a464cb43d2e0f845c5dea2025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912345373456610.3934/math.20241645Composite trapezoidal quadrature for computing hypersingular integrals on intervalXiaoping Zhang0Jin Li1School of Science, Shandong Jianzhu University, Jinan 250101, ChinaSchool of Science, Shandong Jianzhu University, Jinan 250101, ChinaIn this paper, composite trapezoidal quadrature for numerical evaluation of hypersingular integrals was first introduced. By Taylor expansion at the singular point $ y $, error functional was obtained. We know that the divergence rate of $ O(h^{-p}), p = 1, 2 $, and there were no roots of the special function for the first part in the error functional. Meanwhile, for the second part of the error functional, the divergence rate was $ O(h^{-p+1}), p = 1, 2 $, but there were roots of the special function. We proved that the convergence rate could reach $ O(h^{2}) $ at superconvergence points far from the end of the interval. Two modified trapezoidal quadratures are presented and their convergence rate can reach $ O(h^{2}) $ at certain superconvergence points or any local coordinate point. At last, several examples were presented to test our theorem.https://www.aimspress.com/article/doi/10.3934/math.20241645hadamard finite-part integralshypersingular integralscomposite trapezoidal quadratureasymptotic expansionspecial function |
| spellingShingle | Xiaoping Zhang Jin Li Composite trapezoidal quadrature for computing hypersingular integrals on interval AIMS Mathematics hadamard finite-part integrals hypersingular integrals composite trapezoidal quadrature asymptotic expansion special function |
| title | Composite trapezoidal quadrature for computing hypersingular integrals on interval |
| title_full | Composite trapezoidal quadrature for computing hypersingular integrals on interval |
| title_fullStr | Composite trapezoidal quadrature for computing hypersingular integrals on interval |
| title_full_unstemmed | Composite trapezoidal quadrature for computing hypersingular integrals on interval |
| title_short | Composite trapezoidal quadrature for computing hypersingular integrals on interval |
| title_sort | composite trapezoidal quadrature for computing hypersingular integrals on interval |
| topic | hadamard finite-part integrals hypersingular integrals composite trapezoidal quadrature asymptotic expansion special function |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241645 |
| work_keys_str_mv | AT xiaopingzhang compositetrapezoidalquadratureforcomputinghypersingularintegralsoninterval AT jinli compositetrapezoidalquadratureforcomputinghypersingularintegralsoninterval |