{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:18:54Z","timestamp":1760242734313,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2016,4,16]],"date-time":"2016-04-16T00:00:00Z","timestamp":1460764800000},"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>We reply to the comment by Frewer and Khujadze regarding our contribution \u201cLie Symmetry Analysis of the Hopf Functional-Differential Equation\u201d (Symmetry 2015, 7(3), 1536). The method developed by the present authors considered the Lie group analysis of the Hopf equations with functional derivatives in the equation, not the integro-differential equations in general. It was based on previous contributions (Oberlack and Wac\u0142awczyk, Arch. Mech. 2006, 58; Wac\u0142awczyk and Oberlack, J. Math. Phys. 2013, 54). In fact, three of the symmetries calculated in (Symmetry 2015, 7(3), 1536) break due to internal consistency constrains and conditions imposed on test functions, the same concerns the corresponding symmetries derived by Frewer and Khujadze and another, spurious symmetry, which was not discussed by Frewer and Khujadze. As a result, the same set of symmetries is obtained with both approaches.<\/jats:p>","DOI":"10.3390\/sym8040024","type":"journal-article","created":{"date-parts":[[2016,4,18]],"date-time":"2016-04-18T10:37:17Z","timestamp":1460975837000},"page":"24","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Reply to Frewer et al. Comments on Janocha et al. Lie Symmetry Analysis of the Hopf Functional-Differential Equation. Symmetry 2015, 7, 1536\u20131566"],"prefix":"10.3390","volume":"8","author":[{"given":"Marta","family":"Wac\u0142awczyk","sequence":"first","affiliation":[{"name":"Institute of Geophysics, Faculty of Physics, University of Warsaw, Pasteura 7, 02-093 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Janocha","sequence":"additional","affiliation":[{"name":"Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Oberlack","sequence":"additional","affiliation":[{"name":"Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,4,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1536","DOI":"10.3390\/sym7031536","article-title":"Lie symmetry analysis of the Hopf functional-differential equation","volume":"7","author":"Janocha","year":"2015","journal-title":"Symmetry"},{"key":"ref_2","unstructured":"Ibragimov, N.H. (1996). CRC Handbook of Lie Group Analysis of Differential Equations, Volume 3: New Trends in Theoretical Developments and Computational Methods, CRC Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Fushchich, W.I., Shtelen, W.M., and Serov, N.I. (1993). Symmetry Analyis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Springer Verlag.","DOI":"10.1007\/978-94-017-3198-0"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/S0034-4877(01)80088-4","article-title":"Symmetries of integro-differential equations","volume":"48","author":"Zawistowski","year":"2001","journal-title":"Rep. Math. Phys."},{"key":"ref_5","first-page":"597","article-title":"On the extension of Lie group analysis to functional differential equations","volume":"58","author":"Oberlack","year":"2006","journal-title":"Arch. Mech."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Wac\u0142awczyk, M., and Oberlack, M. (2013). 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Lie Symmetry Analysis of the Hopf Functional-Differential Equation. [Master\u2019s Thesis, TU Darmstadt].","DOI":"10.3390\/sym7031536"},{"key":"ref_11","first-page":"87","article-title":"Statistical hydromechanics and functional calculus","volume":"1","author":"Hopf","year":"1952","journal-title":"J. Ration. Mech. Anal."},{"key":"ref_12","unstructured":"Ibragimov, N.H. (1994). CRC Handbook of Lie Group Analysis of Differential Equations, Volume 1: Symmetries, Exact Solutions and Conservation Laws, CRC Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/4\/24\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:22:22Z","timestamp":1760210542000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/4\/24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,4,16]]},"references-count":12,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,4]]}},"alternative-id":["sym8040024"],"URL":"https:\/\/doi.org\/10.3390\/sym8040024","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2016,4,16]]}}}