Lie Symmetry Analysis of the Hopf Functional-Differential Equation

Symmetry

Daniel D. Janocha¹  Marta Wacᐪwczyk¹  Martin Oberlack¹ 

关键词: Lie symmetries;    Hopf equation;    Burgers equation;    functional differential equations;    turbulence;   

DOI  :  10.3390/sym7031536

来源: mdpi

【 摘 要 】

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 productappearing 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.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland.