{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:42:46Z","timestamp":1767339766365,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,7,26]],"date-time":"2022-07-26T00:00:00Z","timestamp":1658793600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","award":["PNURSP2022R52","TURSP-2020\/159","22UQU4282396DSR13"],"award-info":[{"award-number":["PNURSP2022R52","TURSP-2020\/159","22UQU4282396DSR13"]}]},{"name":"Taif University-Saudi Arabia","award":["PNURSP2022R52","TURSP-2020\/159","22UQU4282396DSR13"],"award-info":[{"award-number":["PNURSP2022R52","TURSP-2020\/159","22UQU4282396DSR13"]}]},{"name":"Umm Al-Qura University","award":["PNURSP2022R52","TURSP-2020\/159","22UQU4282396DSR13"],"award-info":[{"award-number":["PNURSP2022R52","TURSP-2020\/159","22UQU4282396DSR13"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The integral equations with oscillatory kernels are of great concern in applied sciences and computational engineering, particularly for large-scale data points and high frequencies. Therefore, the interest of this work is to develop an accurate, efficient, and stable algorithm for the computation of the Fredholm integral equations (FIEs) with the oscillatory kernel. The oscillatory part of the FIEs is evaluated by the Levin quadrature coupled with a compactly supported radial basis function (CS-RBF). The algorithm exhibits sparse and well-conditioned matrix even for large-scale data points, as compared to its counterpart, multi-quadric radial basis function (MQ-RBF) coupled with the Levin quadrature. Usually, the RBFs behave with spherical symmetry about the centers, known as radial. The comparison of convergence and stability analysis of both types of RBFs are performed and numerically verified. The proposed algorithm is tested with benchmark problems and compared with the counterpart methods in the literature. It is concluded that the algorithm in this work is accurate, robust, and stable than the existing methods in the literature based on MQ-RBF and the Chebyshev interpolation matrix.<\/jats:p>","DOI":"10.3390\/sym14081527","type":"journal-article","created":{"date-parts":[[2022,7,29]],"date-time":"2022-07-29T01:41:16Z","timestamp":1659058876000},"page":"1527","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Numerical Investigation of the Fredholm Integral Equations with Oscillatory Kernels Based on Compactly Supported Radial Basis Functions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0069-4025","authenticated-orcid":false,"given":"Suliman","family":"Khan","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Central South University, Changsha 410083, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7761-4196","authenticated-orcid":false,"given":"Sharifah E.","family":"Alhazmi","sequence":"additional","affiliation":[{"name":"Mathematics Department, Al-Qunfudah University College, Umm Al-Qura University, Mecca 24382, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0911-3346","authenticated-orcid":false,"given":"Aisha M.","family":"Alqahtani","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11671, Saudi Arabia"}]},{"given":"Ahmed EI-Sayed","family":"Ahmed","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Taif University, Taif 21944, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3952-4341","authenticated-orcid":false,"given":"Mansour F.","family":"Yaseen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Aflaj, Prince Sattam Bin Abdulaziz University, Al-Aflaj 11912, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3151-9967","authenticated-orcid":false,"given":"Elsayed M.","family":"Tag-Eldin","sequence":"additional","affiliation":[{"name":"Faculty of Engineering and Technology, Future University in Egypt, New Cairo 11835, Egypt"}]},{"given":"Dania","family":"Qaiser","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Central South University, Changsha 410083, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,26]]},"reference":[{"key":"ref_1","unstructured":"Manzhirov, A.V., and Polyanin, A.D. 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