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Symmetry 2016, 8(4), 24; doi:10.3390/sym8040024

Reply to Frewer et al. Comments on Janocha et al. Lie Symmetry Analysis of the Hopf Functional-Differential Equation. Symmetry 2015, 7, 1536–1566

1
Institute of Geophysics, Faculty of Physics, University of Warsaw, Pasteura 7, 02-093 Warsaw, Poland
2
Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Academic Editor: Sergei D. Odintsov
Received: 17 February 2016 / Revised: 9 March 2016 / Accepted: 9 March 2016 / Published: 16 April 2016
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Abstract

We reply to the comment by Frewer and Khujadze regarding our contribution “Lie Symmetry Analysis of the Hopf Functional-Differential Equation” (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 Wacławczyk, Arch. Mech. 2006, 58; Wacławczyk 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. 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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Wacławczyk, M.; Janocha, D.D.; Oberlack, M. Reply to Frewer et al. Comments on Janocha et al. Lie Symmetry Analysis of the Hopf Functional-Differential Equation. Symmetry 2015, 7, 1536–1566. Symmetry 2016, 8, 24.

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