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Symmetry 2015, 7(3), 1536-1566; doi:10.3390/sym7031536

Lie Symmetry Analysis of the Hopf Functional-Differential Equation

Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Str. 2, Darmstadt 64287, Germany
These authors contributed equally to this work.
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Author to whom correspondence should be addressed.
Academic Editor: Sergei D. Odintsov
Received: 17 July 2015 / Revised: 7 August 2015 / Accepted: 20 August 2015 / Published: 27 August 2015
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Abstract

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 Wacławczyk (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. View Full-Text
Keywords: Lie symmetries; Hopf equation; Burgers equation; functional differential equations; turbulence Lie symmetries; Hopf equation; Burgers equation; functional differential equations; turbulence
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Janocha, D.D.; Wacławczyk, M.; Oberlack, M. Lie Symmetry Analysis of the Hopf Functional-Differential Equation. Symmetry 2015, 7, 1536-1566.

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