{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,18]],"date-time":"2025-11-18T09:33:55Z","timestamp":1763458435205,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,12]],"date-time":"2025-05-12T00:00:00Z","timestamp":1747008000000},"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":["KFU251209"],"award-info":[{"award-number":["KFU251209"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This article describes an effective computing method for singularly perturbed parabolic problems with small negative shifts in convection and reaction terms. To handle the small negative shifts, the Taylor series expansion is used. The asymptotically equivalent singularly perturbed parabolic convection\u2013diffusion\u2013reaction problem is then discretized with the Crank\u2013Nicolson method on a uniform mesh for the time derivative and a hybrid method on Shishkin-type meshes for the space derivative. The method\u2019s stability and parameter-uniform convergence are established. To substantiate the theoretical findings, the numerical results are presented in tables and graphs are plotted. The present results improve the existing methods in the literature. Due to the effect of the small negative shifts in Examples 1 and 2, the numerical results using Shishkin and Bakhvalov\u2013Shishkin meshes are almost the same. Since there are no small shifts in Examples 3 and 4, the numerical results using the Bakhvalov\u2013Shishkin mesh are more efficient than using the Shishkin mesh. We conclude that the present method using the Bakhvalov\u2013Shishkin mesh performs well for singularly perturbed problems without small negative shifts.<\/jats:p>","DOI":"10.3390\/axioms14050362","type":"journal-article","created":{"date-parts":[[2025,5,12]],"date-time":"2025-05-12T09:13:38Z","timestamp":1747041218000},"page":"362","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Layer-Adapted Numerical Method for Singularly Perturbed Partial Functional-Differential Equations"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1111-3054","authenticated-orcid":false,"given":"Ahmed A.","family":"Al Ghafli","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\/0000-0001-8169-5294","authenticated-orcid":false,"given":"Hassan J.","family":"Al Salman","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":[[2025,5,12]]},"reference":[{"key":"ref_1","unstructured":"Tzou, D.Y. 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