{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T21:11:52Z","timestamp":1776373912434,"version":"3.51.2"},"reference-count":34,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T00:00:00Z","timestamp":1753142400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["KFU252615"],"award-info":[{"award-number":["KFU252615"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper presents a uniformly convergent cubic spline numerical method for a singularly perturbed two-perturbation parameter ordinary differential equation. The considered differential equation is discretized using the cubic spline numerical method on a Bakhvalov-type mesh. The uniform convergence via the error analysis is established very well. The numerical findings indicate that the proposed method achieves second-order uniform convergence. Four test examples have been considered to perform numerical experimentations.<\/jats:p>","DOI":"10.3390\/axioms14080547","type":"journal-article","created":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T08:45:18Z","timestamp":1753173918000},"page":"547","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Cubic Spline Numerical Method for a Singularly Perturbed Two-Parameter Ordinary Differential Equation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8169-5294","authenticated-orcid":false,"given":"Hassan J.","family":"Al Salman","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf 319832, Al Ahsa, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4693-9051","authenticated-orcid":false,"given":"Fasika Wondimu","family":"Gelu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Natural and Computational Sciences, Dilla University, Dilla 419, Ethiopia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1111-3054","authenticated-orcid":false,"given":"Ahmed A.","family":"Al Ghafli","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf 319832, Al Ahsa, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/0022-247X(67)90124-2","article-title":"Singular perturbations of boundary value problems for linear ordinary differential equations involving two parameters","volume":"19","year":"1967","journal-title":"J. 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