{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:45:27Z","timestamp":1747190727147,"version":"3.40.5"},"reference-count":22,"publisher":"Wiley","license":[{"start":{"date-parts":[[2021,10,16]],"date-time":"2021-10-16T00:00:00Z","timestamp":1634342400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Scientific Programming"],"published-print":{"date-parts":[[2021,10,16]]},"abstract":"<jats:p>Symmetry methods for differential equations are a powerful tool for the solutions of differential equations. It linearizes nonlinear differential equations, reduces the order of differential equations, reduces the number of independent variables in partial differential equations, and solves almost all those differential equations for which the other analytic methods fail to solve them. Similarity transformation is a particular case of symmetries, but it is easy and often used to deal with differential equations. The similarity transformation can do all the aforementioned works. In this research, we use the similarity transformation to solve different nonlinear differential equations. Particularly, we will apply this transformation to the nonlinear Navier\u2013Stokes partial differential equations to reduce them to ordinary differential equations. Ordinary differential equations are easy to deal with than partial differential equations. Some nonlinear physical examples of ODEs and PDEs are given to show that the similarity transformation solves those problems where the other analytic methods fail to work.<\/jats:p>","DOI":"10.1155\/2021\/9054328","type":"journal-article","created":{"date-parts":[[2021,10,16]],"date-time":"2021-10-16T18:20:51Z","timestamp":1634408451000},"page":"1-11","source":"Crossref","is-referenced-by-count":0,"title":["Reduction of Asymptotic Approximate Expansion of Navier\u2013Stokes Equation and Solution of Inviscid Burgers Equation by Similarity Transformation"],"prefix":"10.1155","volume":"2021","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7012-3597","authenticated-orcid":true,"given":"Farhad","family":"Ali","sequence":"first","affiliation":[{"name":"Institute of Numerical Sciences, Kohat University of Sciences and Technology, Kohat, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5081-741X","authenticated-orcid":true,"given":"Wali Khan","family":"Mashwani","sequence":"additional","affiliation":[{"name":"Institute of Numerical Sciences, Kohat University of Sciences and Technology, Kohat, Pakistan"}]},{"given":"Hamayat","family":"Ullah","sequence":"additional","affiliation":[{"name":"Institute of Numerical Sciences, Kohat University of Sciences and Technology, Kohat, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0186-0526","authenticated-orcid":true,"given":"Ahmed","family":"Hussein Msmali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Collage of Science, Jazan University, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2366-8041","authenticated-orcid":true,"given":"Ikramullah","family":"Ikramullah","sequence":"additional","affiliation":[{"name":"Department of Physics, Kohat University of Sciences and Technology, Kohat, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5877-9051","authenticated-orcid":true,"given":"Zabidin","family":"Salleh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1021\/ie0510425"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/s0898-1221(01)00312-1"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247x(84)90182-3"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.26422\/mat.a.2000.2.tar"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.2298\/tsci140326074h"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/j.compchemeng.2017.02.047"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1515\/phys-2020-0210"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1137\/060671991"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2003.09.024"},{"key":"10","article-title":"Similarity transformation methods in the analysis of the two dimensional steady compressible laminar boundary layer","volume":"2","author":"Y. 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