Generalization of the Equivalence Transformations

Meleshko, Sergey V.

doi:10.2991/jnmp.1996.3.1-2.21

Cited by 35 publications

(47 citation statements)

References 11 publications

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“…More in general, taking the transformations of the independent and dependent variables as functions of the arbitrary elements p too [71], in the expression of prolongation we have to replace the Lie derivative D Dx i with…”

Section: Equivalence Transformationsmentioning

confidence: 99%

“…Remark 7 By default, the infinitesimals for the independent and dependent variables do not depend on arbitrary elements; if we are interested to general equivalence transformations where also the infinitesimal generators of the independent and dependent variables depend on the arbitrary elements [71], then we have to add the statement generalequiv:=1 $…”

Section: Computation Of Equivalence Transformationsmentioning

confidence: 99%

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ReLie: a Reduce program for Lie group analysis of differential equations

Oliveri¹

2021

Preprint

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Lie symmetry analysis provides a general theoretical framework for investigating ordinary and partial differential equations. The theory is completely algorithmic even if it usually involves lengthy computations. For this reason, many computer algebra packages have been developed along the years to automate the computation. In this paper, we describe the program ReLie, written in the Computer Algebra System Reduce, which since 2008 is an open source program (http://www.reduce-algebra.com) and is available for all platforms. ReLie is able to perform almost automatically the needed computations for Lie symmetry analysis of differential equations. Its source code is freely available at the url http://mat521.unime.it/oliveri. The use of the program is illustrated by means of some simple examples; nevertheless, it is to be underlined that it provides effective also for more complex computations where one has to deal with very large expressions.

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Section: Equivalence Transformationsmentioning

confidence: 99%

Section: Computation Of Equivalence Transformationsmentioning

confidence: 99%

ReLie: a Reduce program for Lie group analysis of differential equations

Oliveri¹

2021

Preprint

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“…Then these equivalence transformations constitute a Lie pseudogroup Ḡ∼ called the generalized equivalence group of the class. See the first discussion of this notion in [3,4] and the further development in [8,9]. When the generalized equivalence group coincides with the usual one the situation is considered to be trivial.…”

Section: Equivalence Of Classes Of Differential Equationsmentioning

confidence: 99%

Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients

Opanasenko

2019

Preprint

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We describe the equivalence groupoid of the class of general Burgers -Korteweg -de Vries equations with space-dependent coefficients. This class is shown to reduce by a family of equivalence transformations to a subclass whose usual equivalence group is four-dimensional. Classified are admissible transformations of this subclass and singled out its subclasses admitting maximal nontrivial conditional equivalence groups. All of them turn out to have dimension higher than four. In particular, a few new examples of nontrivial cases of normalization in the generalized sense of classes of differential equations appeared this way.

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“…Needless to say, that in general not all admissible transformations are induced by elements from the equivalence group. Different generalizations of the notion of usual equivalence groups exist in the literature [30,40]. By g ∼ we denote the algebra associated with the equivalence group G ∼ and call it the equivalence algebra of the class L| S .…”

Section: Parameterization Via Direct Group Classificationmentioning

confidence: 99%

“…As we compute the usual equivalence algebra rather than the generalized one [30] and the arbitrary elements f i do not depend on fourth order derivatives of ψ, the elements of the algebra are assumed to be vector fields in the joint space of the variables of J (3) and the arbitrary elements f i , which are projectable to both the spaces (t, x, y, ψ) and J (3) . In other words, the algebra consists of vector fields of the general form…”

Section: Equivalence Algebras Of Classes Of Generalized Vorticity Equ...mentioning

confidence: 99%

Symmetry preserving parameterization schemes

Popovych

Bihlo

2012

Journal of Mathematical Physics

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Methods for the design of physical parameterization schemes that possess certain invariance properties are discussed. These methods are based on different techniques of group classification and provide means to determine expressions for unclosed terms arising in the course of averaging of nonlinear differential equations. The demand that the averaged equation is invariant with respect to a subalgebra of the maximal Lie invariance algebra of the unaveraged equation leads to a problem of inverse group classification which is solved by the description of differential invariants of the selected subalgebra. Given no prescribed symmetry group, the direct group classification problem is relevant. Within this framework, the algebraic method or direct integration of determining equations for Lie symmetries can be applied. For cumbersome parameterizations, a preliminary group classification can be carried out. The methods presented are exemplified by parameterizing the eddy vorticity flux in the averaged vorticity equation. In particular, differential invariants of (infinite dimensional) subalgebras of the maximal Lie invariance algebra of the unaveraged vorticity equation are computed. A hierarchy of normalized subclasses of generalized vorticity equations is constructed. Invariant parameterizations possessing minimal symmetry extensions are described and a restricted class of invariant parameterization is exhaustively classified. The physical importance of the parameterizations designed is discussed.

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Generalization of the Equivalence Transformations

Cited by 35 publications

References 11 publications

ReLie: a Reduce program for Lie group analysis of differential equations

ReLie: a Reduce program for Lie group analysis of differential equations

Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients

Symmetry preserving parameterization schemes

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