A consistent approach to approximate Lie symmetries of differential equations

Abstract

Lie theory of continuous transformations provides a unified and powerful approach for handling differential equations. Unfortunately, any small perturbation of an equation usually destroys some important symmetries, and this reduces the applicability of Lie group methods to differential equations arising in concrete applications. On the other hand, differential equations containing small terms are commonly and successfully investigated by means of perturbative techniques. Therefore, it is desirable to combine Lie group methods with perturbation analysis, i.e., to establish an approximate symmetry theory. There are two widely used approaches to approximate symmetries: the one proposed in 1988 by Baikov, Gazizov and Ibragimov, and the one introduced in 1989 by Fushchich and Shtelen. Moreover, some variations of the Fushchich-Shtelen method have been proposed with the aim of reducing the length of computations. Here, we propose a new approach that is consistent with perturbation theory and allows to extend all the relevant features of Lie group analysis to an approximate context. Some applications are also presented.

Approximately invariant solutions of creeping flow equations

“…In this paper, we explicitly determine some classes of approximately invariant solutions of the steady creeping flow equations of second grade fluids by using a recently introduced approach [28] to approximate Lie symmetries that is consistent with the principles of perturbative analysis. The same equations have been analyzed in [16], where the results of three different approximate symmetry methods have been compared.…”

“…and using the consistent approach to approximate Lie symmetries [28], we obtain that equations (22) are approximately (at first order) invariant with respect to the approximate Lie groups of point transformations generated by the following vector fields:…”

“…In this paper, we apply this consistent approach to the creeping flow equations of a second grade fluid; the choice is motivated by the fact that this model has been analyzed in [16] by means of the different known approaches to approximate Lie symmetries whose results have been compared. Our aim is to show that the new consistent approach can effectively be applied in a concrete situation producing reliable results; the method proposed in [28] combines the ideas underlying the various approaches thus allowing to take into account the principles of invariance under a continuous group of transformations and perturbative analysis, so that it seems the more adequate one to deal with differential equations containing small terms.…”

Tremor Estimation and Removal in Robot-Assisted Surgery Using Lie Groups and EKF

“…In ref. [18], the theory of approximate group Lie symmetries for different equations has been developed. Given a differential equation, it may admit many point symmetries defined by Lie algebra generators and such symmetries enable us to find solutions of the differential equation easily.…”

ReLie: A Reduce Program for Lie Group Analysis of Differential Equations

“…Among these generalizations there are the nonclassical symmetries first proposed by Bluman and Cole [43], and now part of the more general method of differential constraints [44,45], the potential symmetries [46], the nonlocal symmetries [47][48][49], the gen-eralized symmetries [5], which in turn generalize contact symmetries introduced by Lie himself, the equivalence transformations [3,[50][51][52][53][54][55][56], to quote a few. A further extension is represented by approximate symmetries [57][58][59][60][61] for differential equations containing small terms, often arising in concrete applied problems (see, for instance, [62] for an application of approximate Lie symmetries to Navier-Stokes equations).…”