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

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.

Anisotropic stars in modified gravity: An extended gravitational decoupling approach*

“…It is well known that the n dimensional manifold can always be embedded in Pseudo-Euclidean space of dimensions, and the minimum extra dimensions of pseudo-Euclidean space needed for the embedding is called the embedding Class of . According to this definition, Schwarzschild 's interior and exterior solutions are of Class I and Class II, respectively; the class of general spherical and plane symmetric spacetimes are II and III, respectively; and the well known Kerr metric is of Class V [62][63][64][65].…”