{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:23:52Z","timestamp":1760243032283,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2015,8,27]],"date-time":"2015-08-27T00:00:00Z","timestamp":1440633600000},"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 paper, we extend the classical Lie symmetry analysis from partial differential equations to integro-differential equations with functional derivatives. We continue the work of Oberlack and Wac\u0142awczyk (2006, Arch. Mech. 58, 597), (2013, J. Math. Phys. 54, 072901), where the extended Lie symmetry analysis is performed in the Fourier space. Here, we introduce a method to perform the extended Lie symmetry analysis in the physical space where we have to deal with the transformation of the integration variable in the appearing integral terms. The method is based on the transformation of the product y(x)dx appearing in the integral terms and applied to the functional formulation of the viscous Burgers equation. The extended Lie symmetry analysis furnishes all known symmetries of the viscous Burgers equation and is able to provide new symmetries associated with the Hopf formulation of the viscous Burgers equation. Hence, it can be employed as an important tool for applications in continuum mechanics.<\/jats:p>","DOI":"10.3390\/sym7031536","type":"journal-article","created":{"date-parts":[[2015,8,27]],"date-time":"2015-08-27T10:03:37Z","timestamp":1440669817000},"page":"1536-1566","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Lie Symmetry Analysis of the Hopf Functional-Differential Equation"],"prefix":"10.3390","volume":"7","author":[{"given":"Daniel","family":"Janocha","sequence":"first","affiliation":[{"name":"Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Str. 2, Darmstadt 64287, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marta","family":"Wac\u0142awczyk","sequence":"additional","affiliation":[{"name":"Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Str. 2, Darmstadt 64287, 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, Darmstadt 64287, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,8,27]]},"reference":[{"key":"ref_1","first-page":"451","article-title":"New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws","volume":"3","author":"Oberlack","year":"2010","journal-title":"Disc. Cont. Dyn. Sys. Ser. S"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1063\/1.1762249","article-title":"Distribution functions in the statistical theory of turbulence","volume":"10","author":"Lundgren","year":"1967","journal-title":"Phys. Fluids"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wac\u0142awczyk, M., Staffolani, N., Oberlack, M., Rosteck, A., Wilczek, M., and Friedrich, R. (2014). Statistical Symmetries of the Lundgren\u2013Monin\u2013Novikov Hierarchy. Phys. Rev. E, 90.","DOI":"10.1103\/PhysRevE.90.013022"},{"key":"ref_4","first-page":"87","article-title":"Statistical Hydromechanics and Functional Calculus","volume":"1","author":"Hopf","year":"1952","journal-title":"J. Ration. Mech. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hosokawa, I. (2006). Monin-Lundgren hierarchy versus the Hopf equation in the statistical theory of turbulence. Phys. Rev. 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CRC Handbook of Lie Group Analysis of Differential Equations, Volume 1: Symmetries, Exact Solutions and Conservation Laws, CRC Press."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Klauder, J.R. (2011). A Modern Approach to Functional Integration, Birkh\u00e4user.","DOI":"10.1007\/978-0-8176-4791-9"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1023\/A:1015061100660","article-title":"Symmetries of integro-differential equations: A survey of methods illustrated by the Benny equations","volume":"28","author":"Ibragimov","year":"2002","journal-title":"Nonlinear Dyn."},{"key":"ref_16","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_17","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_18","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/j.mechrescom.2004.10.002","article-title":"Symmetry group analysis of Benney system and an application for shallow-water equations","volume":"32","year":"2005","journal-title":"Mech. Res. Commun."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Oberlack, M., Wac\u0142awczyk, M., Rosteck, A., and Avsarkisov, V. (2015). Symmetries and their importance for statistical turbulence theory. Mech. Eng. 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