{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T02:26:55Z","timestamp":1772504815897,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T00:00:00Z","timestamp":1725840000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2230790"],"award-info":[{"award-number":["DMS-2230790"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2212938"],"award-info":[{"award-number":["DMS-2212938"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2309491"],"award-info":[{"award-number":["DMS-2309491"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Nonstandard finite-difference (NSFD) methods, pioneered by R. E. Mickens, offer accurate and efficient solutions to various differential equation models in science and engineering. NSFD methods avoid numerical instabilities for large time steps, while numerically preserving important properties of exact solutions. However, most NSFD methods are only first-order accurate. This paper introduces two new classes of explicit second-order modified NSFD methods for solving n-dimensional autonomous dynamical systems. These explicit methods extend previous work by incorporating novel denominator functions to ensure both elementary stability and second-order accuracy. This paper also provides a detailed mathematical analysis and validates the methods through numerical simulations on various biological systems.<\/jats:p>","DOI":"10.3390\/computation12090183","type":"journal-article","created":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T11:31:59Z","timestamp":1725881519000},"page":"183","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Second-Order Modified Nonstandard Explicit Euler and Explicit Runge\u2013Kutta Methods for n-Dimensional Autonomous Differential Equations"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-1681-7329","authenticated-orcid":false,"given":"Fawaz K.","family":"Alalhareth","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Arts and Sciences, Najran University, Najran 55461, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7668-3114","authenticated-orcid":false,"given":"Madhu","family":"Gupta","sequence":"additional","affiliation":[{"name":"Department of Computing and Data Science, Flame University, Pune 412115, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5638-5427","authenticated-orcid":false,"given":"Hristo V.","family":"Kojouharov","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The University of Texas at Arlington, Arlington, TX 76019, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6134-1381","authenticated-orcid":false,"given":"Souvik","family":"Roy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The University of Texas at Arlington, Arlington, TX 76019, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mickens, R. 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