{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T02:56:04Z","timestamp":1775012164174,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,22]],"date-time":"2021-01-22T00:00:00Z","timestamp":1611273600000},"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>The Lie symmetry group transformation method was used to investigate the partial differential equations that model the motion of a natural convective unsteady flow past to a non-isothermal vertical flat surface. The one-parameter Lie group transformation was applied twice consecutively to convert the motion governing equations into a system of ordinary differential equations. The obtained system of ordinary differential equations was solved numerically using the Lobatto IIIA formula (implicit Runge\u2013Kutta method). The effect of the Prandtl number on the temperature and velocity profiles is illustrated graphically.<\/jats:p>","DOI":"10.3390\/sym13020175","type":"journal-article","created":{"date-parts":[[2021,1,25]],"date-time":"2021-01-25T12:28:31Z","timestamp":1611577711000},"page":"175","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Lie Symmetry Group for Unsteady Free Convection Boundary-Layer Flow over a Vertical Surface"],"prefix":"10.3390","volume":"13","author":[{"given":"Mina B.","family":"Abd-el-Malek","sequence":"first","affiliation":[{"name":"Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nagwa A.","family":"Badran","sequence":"additional","affiliation":[{"name":"Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amr M.","family":"Amin","sequence":"additional","affiliation":[{"name":"Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anood M.","family":"Hanafy","sequence":"additional","affiliation":[{"name":"Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,22]]},"reference":[{"key":"ref_1","unstructured":"Oleinik, O.A., and Samokhin, V.N. (1999). Mathematical Models in Boundary Layer Theory, Chapman& Hall\/CRC. [1st ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1063\/1.2169443","article-title":"Ludwig Prandtl\u2019s boundary layer","volume":"58","author":"Anderson","year":"2005","journal-title":"Phys. Today"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Steeb, W.H. (1993). Invertible Point Transformations and Nonlinear Differential Equations, World Scientific. [1st ed.].","DOI":"10.1142\/1987"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hydon, P.E. (2000). Symmetry Methods for Differential Equations, Cambridge University Press.","DOI":"10.1017\/CBO9780511623967"},{"key":"ref_5","unstructured":"Hundley, J. (2009). Introduction to Lie Groups, Massachusetts Institute of Technology. Miscellaneous."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hall, B.C. (2013). Lie groups, Lie algebras, and representations. 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Ed."},{"key":"ref_14","first-page":"132","article-title":"Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption\/generation and suction\/blowing: A lie group analysis","volume":"7","author":"Hamad","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer Simulat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1338","DOI":"10.1016\/j.nonrwa.2010.09.014","article-title":"Magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate","volume":"12","author":"Hamad","year":"2011","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"239082","DOI":"10.1155\/2014\/239082","article-title":"Lie group solution for free convective flow of a nanofluid past a chemically reacting horizontal plate in porous media","volume":"2014","author":"Rashidi","year":"2014","journal-title":"Math. Probl. 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