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Sibirsk. Mat. Zh., 2013, Volume 54, Number 6, Pages 1396–1406 (Mi smj2505)  

This article is cited in 7 scientific papers (total in 7 papers)

A hierarchy of submodels of differential equations

S. V. Khabirovab

a Institute of Mechanics, Ufa Centre of the Russian Academy of Sciences, Ufa, Russia
b Ufa State Aviation Technical University, Ufa, Russia

Abstract: We consider a system of differential equations admitting a group of transformations. The Lie algebra of the group generates a hierarchy of submodels. This hierarchy can be chosen so that the solutions to each of submodels are solutions to some other submodel in the same hierarchy. For this we must calculate an optimal system of subalgebras and construct a graph of embedded subalgebras and then calculate the differential invariants and invariant differentiation operators for each subalgebra. The invariants of a superalgebra are functions of the invariants of the algebra. The invariant differentiation operators of a superalgebra are linear combinations of invariant differentiation operators of a subalgebra over the field of invariants of the subalgebra. The comparison of the representations of group solutions gives a relation between the solutions to the models of the superalgebra and the subalgebra. Some examples are given of embedded submodels for the equations of gas dynamics.

Keywords: differential invariant submodels, hierarchy of submodels, gas dynamics.

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English version:
Siberian Mathematical Journal, 2013, 54:6, 1110–1119

Bibliographic databases:

UDC: 517.958
Received: 07.02.2012
Revised: 21.12.2012

Citation: S. V. Khabirov, “A hierarchy of submodels of differential equations”, Sibirsk. Mat. Zh., 54:6 (2013), 1396–1406; Siberian Math. J., 54:6 (2013), 1110–1119

Citation in format AMSBIB
\by S.~V.~Khabirov
\paper A hierarchy of submodels of differential equations
\jour Sibirsk. Mat. Zh.
\yr 2013
\vol 54
\issue 6
\pages 1396--1406
\jour Siberian Math. J.
\yr 2013
\vol 54
\issue 6
\pages 1110--1119

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    This publication is cited in the following articles:
    1. R. F. Shayakhmetova, “Vlozhennye invariantnye podmodeli dvizheniya odnoatomnogo gaza”, Sib. elektron. matem. izv., 11 (2014), 605–625  mathnet
    2. S. V. Khabirov, “Group analysis of a one-dimensional model of gas flow in a porous medium”, J. Appl. Math. Mech., 81:4 (2017), 334–340  crossref  mathscinet  isi  elib
    3. S. V. Khabirov, R. F. Shayakhmetova, “Prostye resheniya invariantnoi podmodeli ranga 2 odnoatomnogo gaza”, Chelyab. fiz.-matem. zhurn., 3:3 (2018), 353–373  mathnet  crossref  elib
    4. R. F. Nikonorova, “Podmodeli odnoatomnogo gaza naimenshego ranga, postroennye na osnove trekhmernykh podalgebr simmetrii”, Sib. elektron. matem. izv., 15 (2018), 1216–1226  mathnet  crossref
    5. T. F. Mukminov, S. V. Khabirov, “Graf vlozhennykh podalgebr 11-mernoi algebry simmetrii sploshnoi sredy”, Sib. elektron. matem. izv., 16 (2019), 121–143  mathnet  crossref
    6. S. V. Khabirov, “Simple partially invariant solutions”, Ufa Math. J., 11:1 (2019), 90–99  mathnet  crossref  isi
    7. A. A. Gainetdinova, R. K. Gazizov, “Integration of systems of two second-order ordinary differential equations with a small parameter that admit four essential operators”, Sib. elektron. matem. izv., 17 (2020), 604–614  mathnet  crossref
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