{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:38:31Z","timestamp":1760060311699,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,18]],"date-time":"2025-08-18T00:00:00Z","timestamp":1755475200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan","award":["BR27100483"],"award-info":[{"award-number":["BR27100483"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>This research presents a comparative analysis of numerical methods for solving first-kind Fredholm integral equations using the Bubnov\u2013Galerkin method with various wavelet and orthogonal polynomial bases. The bases considered are constructed from Legendre, Laguerre, Chebyshev, and Hermite wavelets, as well as Alpert multiwavelets and CAS wavelets. The effectiveness of these bases is evaluated by measuring errors relative to known analytical solutions at different discretization levels. Results show that global orthogonal systems\u2014particularly the Chebyshev and Hermite\u2014achieve the lowest error norms for smooth target functions. CAS wavelets, due to their localized and oscillatory nature, produce higher errors, though their accuracy improves with finer discretization. The analysis has been extended to incorporate perturbations in the form of additive noise, enabling a rigorous assessment of the method\u2019s stability with respect to different wavelet bases. This approach provides insight into the robustness of the numerical scheme under data uncertainty and highlights the sensitivity of each basis to noise-induced errors.<\/jats:p>","DOI":"10.3390\/computation13080199","type":"journal-article","created":{"date-parts":[[2025,8,18]],"date-time":"2025-08-18T11:31:37Z","timestamp":1755516697000},"page":"199","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Comparative Analysis of Wavelet Bases for Solving First-Kind Fredholm Integral Equations"],"prefix":"10.3390","volume":"13","author":[{"given":"Nurlan","family":"Temirbekov","sequence":"first","affiliation":[{"name":"Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan"},{"name":"National Engineering Academy of the Republic of Kazakhstan, Almaty 050060, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8315-5849","authenticated-orcid":false,"given":"Dinara","family":"Tamabay","sequence":"additional","affiliation":[{"name":"Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan"},{"name":"National Engineering Academy of the Republic of Kazakhstan, Almaty 050060, Kazakhstan"}]},{"given":"Aigerim","family":"Tleulesova","sequence":"additional","affiliation":[{"name":"Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan"},{"name":"National Engineering Academy of the Republic of Kazakhstan, Almaty 050060, Kazakhstan"}]},{"given":"Tomiris","family":"Mukhanova","sequence":"additional","affiliation":[{"name":"Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan"},{"name":"National Engineering Academy of the Republic of Kazakhstan, Almaty 050060, Kazakhstan"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1515\/jiip-2019-5001","article-title":"Theory and Numerical Methods for Solving Inverse and Ill-Posed Problems","volume":"27","author":"Kabanikhin","year":"2019","journal-title":"J. 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