Reduction of Balance Laws in (3+1)-Dimensions to Autonomous Conservation Laws by Means of Equivalence Transformations

Gorgone, Matteo; Oliveri, Francesco; Speciale, Maria Paola

doi:10.1007/s10440-014-9929-5

Cited by 5 publications

(3 citation statements)

References 33 publications

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“…Further applications for deriving approximate conservation laws, or local transformations (suggested by the approximate symmetries) mapping differential equations to approximately equivalent ones are possible. Moreover, either approximate equivalence transformations [1,58,59,60] for classes of differential equations involving small terms or approximate conditional symmetries can be defined [61,62,63]. Some of these extensions are currently under investigation.…”

Section: Discussionmentioning

confidence: 99%

A consistent approach to approximate Lie symmetries of differential equations

2017

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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.

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Section: Discussionmentioning

confidence: 99%

A consistent approach to approximate Lie symmetries of differential equations

2017

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“…When the infinitesimals ξ i and η α are assumed to be independent of p, it is possible to project the symmetries on the space Z ≡ X × U of the independent and dependent variables, so obtaining a transformation changing an element of the class of differential equations to another element in the same class (see [115][116][117] for some applications of these ideas).…”

Section: Equivalence Transformationsmentioning

confidence: 99%

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

Oliveri

2021

Symmetry

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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, along the years many computer algebra packages have been developed to automate the computation. In this paper, we describe the program ReLie, written in the Computer Algebra System Reduce, since 2008 an open source program 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 too. The use of the program is illustrated by means of some examples; nevertheless, it is to be underlined that it proves effective also for more complex computations where one has to deal with very large expressions.

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“…If we project the symmetries on the space Z ≡ X × U of the independent and dependent variables (this is always possible if the infinitesimals of independent and dependent variables are assumed to be independent of p), we obtain a transformation changing an element of the class of differential equations to another element in the same class (same differential structure but in general different arbitrary elements) [85,86,44]. Such projected transformations map solutions of a system in the class to solutions of a transformed system in the same class.…”

Section: Equivalence Transformationsmentioning

confidence: 99%

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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Reduction of Balance Laws in (3+1)-Dimensions to Autonomous Conservation Laws by Means of Equivalence Transformations

Cited by 5 publications

References 33 publications

A consistent approach to approximate Lie symmetries of differential equations

A consistent approach to approximate Lie symmetries of differential equations

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

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

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