{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,25]],"date-time":"2026-01-25T05:07:39Z","timestamp":1769317659275,"version":"3.49.0"},"reference-count":14,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,11]],"date-time":"2024-09-11T00:00:00Z","timestamp":1726012800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this study, we investigate the symmetry analysis and explicit solutions for the Estevez\u2013Mansfield\u2013Clarkson (EMC) equation. Our main objectives are to identify the Lie point symmetries of the EMC equation, construct an optimal system of one-dimensional subalgebras, and reduce the EMC equation to a set of ordinary differential equations (ODEs). We employ the Riccati\u2013Bernoulli sub-ODE method (RBSODE) to solve these reduced ODEs and obtain explicit solutions for the EMC model. The obtained solutions are validated using numerical analyses, and corresponding figures are presented to illustrate the physical implications of the derived solutions.<\/jats:p>","DOI":"10.3390\/sym16091194","type":"journal-article","created":{"date-parts":[[2024,9,11]],"date-time":"2024-09-11T05:45:12Z","timestamp":1726033512000},"page":"1194","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Lie Symmetry Analysis and Explicit Solutions to the Estevez\u2013Mansfield\u2013Clarkson Equation"],"prefix":"10.3390","volume":"16","author":[{"given":"Aliyu Isa","family":"Aliyu","sequence":"first","affiliation":[{"name":"Operational Research Centre in Healthcare, Near East University, TRNC Mersin 10, Nicosia 99138, Turkey"},{"name":"Department of Mathematics, Federal University Dutse, Dutse 720222, Nigeria"},{"name":"School of Computing, Maryam Abacha American University of Nigeria, Kano 700223, Nigeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-0170-3466","authenticated-orcid":false,"given":"Jibrin Sale","family":"Yusuf","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Federal University Dutse, Dutse 720222, Nigeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3489-6792","authenticated-orcid":false,"given":"Malik Muhammad","family":"Nauman","sequence":"additional","affiliation":[{"name":"Faculty of Integrated Technologies, Universiti Brunei Darussalam, Jalan Tungku Link, Gadong, Bandar Seri Begawan BE1410, Brunei"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dilber Uzun","family":"Ozsahin","sequence":"additional","affiliation":[{"name":"Operational Research Centre in Healthcare, Near East University, TRNC Mersin 10, Nicosia 99138, Turkey"},{"name":"Medical Diagnostic Imaging, College of Health Science, University of Sharjah, Sharjah 27272, United Arab Emirates"},{"name":"Research Institute for Medical and Health Sciences, University of Sharjah, Sharjah 27272, United Arab Emirates"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Baba Galadima","family":"Agaie","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Federal University Dutse, Dutse 720222, Nigeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7096-7573","authenticated-orcid":false,"given":"Juliana Haji","family":"Zaini","sequence":"additional","affiliation":[{"name":"Faculty of Integrated Technologies, Universiti Brunei Darussalam, Jalan Tungku Link, Gadong, Bandar Seri Begawan BE1410, Brunei"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2508-9710","authenticated-orcid":false,"given":"Huzaifa","family":"Umar","sequence":"additional","affiliation":[{"name":"Operational Research Centre in Healthcare, Near East University, TRNC Mersin 10, Nicosia 99138, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,11]]},"reference":[{"key":"ref_1","unstructured":"Whitham, G.B. (1974). Linear and Nonlinear Waves, John Whiley."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hesegawa, A., and Kodama, Y. (1995). Solitons in Optical Communication, Oxford University Press.","DOI":"10.1093\/oso\/9780198565079.001.0001"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Yang, X.F., Deng, Z.C., and Wei, Y.A. (2015). A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Adv. Differ. Equ., 117.","DOI":"10.1186\/s13662-015-0452-4"},{"key":"ref_4","first-page":"367","article-title":"Riccati-Bernoulli Sub-ODE approach on the partial differential equations and applications","volume":"15","author":"Alharbi","year":"2020","journal-title":"Int. J. Math. Comput. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Aliyu, A.I., Mustafa Inc., Yusuf, A., and Baleanu, D. (2018). Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Sixth-Order Nonlinear Ramani Equation. 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Power Res."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/9\/1194\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:53:44Z","timestamp":1760111624000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/9\/1194"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,11]]},"references-count":14,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["sym16091194"],"URL":"https:\/\/doi.org\/10.3390\/sym16091194","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,11]]}}}