{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:16:18Z","timestamp":1760145378055,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T00:00:00Z","timestamp":1720656000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this article, we study the semi-linear two-dimensional reaction\u2013diffusion equation with Dirichlet boundaries. A reliable numerical scheme is designed, coupling the nonstandard finite difference method in the time together with the Galerkin in combination with the compactness method in the space variables. The aforementioned equation is analyzed to show that the weak or variational solution exists uniquely in specified space. The a priori estimate obtained from the existence of the weak or variational solution is used to show that the designed scheme is stable and converges optimally in specified norms. Furthermore, we show that the scheme preserves the qualitative properties of the exact solution. Numerical experiments are presented with a carefully chosen example to validate our proposed theory.<\/jats:p>","DOI":"10.3390\/computation12070142","type":"journal-article","created":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T11:30:55Z","timestamp":1720697455000},"page":"142","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Theory and Computation of the Semi-Linear Reaction\u2013Diffusion Equation with Dirichlet Boundaries"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6336-4122","authenticated-orcid":false,"given":"Pius W. M.","family":"Chin","sequence":"first","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa, Ga-Rankuwa, Pretoria 0204, South Africa"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,11]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Study of a diffusion equation, that is related to the growth of a quality of matter and its application to a biological problem","volume":"1","author":"Kolmogorov","year":"1937","journal-title":"Mosc. Univ. Math. Bull."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1017\/S0022112069000176","article-title":"Finite bandwidth, finite amplitude convection","volume":"38","author":"Whitehead","year":"1969","journal-title":"J. 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