{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T03:42:12Z","timestamp":1776397332352,"version":"3.51.2"},"reference-count":23,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2025,3,4]],"date-time":"2025-03-04T00:00:00Z","timestamp":1741046400000},"content-version":"vor","delay-in-days":62,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2025,1]]},"abstract":"<jats:p>\n                    In the present work, a class of singularly perturbed unsteady reaction\u2013diffusion problem is considered. With the existence of a small parameter\n                    <jats:italic>\u03b5<\/jats:italic>\n                    , (0 &lt;\n                    <jats:italic>\u03b5<\/jats:italic>\n                    \u226a 1) as a coefficient of the diffusion term in the proposed model problem, there exist twin boundary layer regions near the left end point\n                    <jats:italic>x<\/jats:italic>\n                    = 0 and right end point\n                    <jats:italic>x<\/jats:italic>\n                    = 1 of the spatial domain. The solutions found in such regions have abrupt changes if not fine meshes are used during the spatial domain discretization. Due to these reasons, the classical numerical methods are inefficient to overcome these challenges. To address the suggested problem, we developed and examined a uniformly convergent numerical scheme. The Crank\u2013Nicolson approach for the temporal direction and nonstandard finite difference method for the spatial direction on uniform meshes are used to discretize the continuous problem domain. The uniform convergence analysis shows that the proposed scheme is second\u2010order uniformly convergent in both temporal and spatial dimensions. Three model examples were given for simulation in order to support the reliability of the formulated scheme, and the obtained results verified that the theoretical analyses coincide with the practical examples. Further, the acquired numerical results demonstrate that the present approach performs better than some of the existing methods in the literature.\n                  <\/jats:p>","DOI":"10.1155\/jama\/4603501","type":"journal-article","created":{"date-parts":[[2025,3,11]],"date-time":"2025-03-11T22:32:12Z","timestamp":1741732332000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Uniformly Convergent Scheme for Singularly Perturbed Unsteady Reaction\u2013Diffusion Problems"],"prefix":"10.1155","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5866-8722","authenticated-orcid":false,"given":"Amare Worku","family":"Demsie","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2548-3278","authenticated-orcid":false,"given":"Awoke Andargie","family":"Tiruneh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7526-3003","authenticated-orcid":false,"given":"Endalew Getnet","family":"Tsega","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,3,4]]},"reference":[{"key":"e_1_2_11_1_2","volume-title":"Robust Numerical Methods for Singularly Perturbed Differential Equations","author":"Roos H. 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