Abstract:
The algorithm for the order reduction of ordinary differential equations (ODEs) by using the operator of invariant differentiation (OID) of admitted Lie algebra is modified for systems of ODEs with a small parameter that admit approximate Lie algebras of operators. Invariant representations of second-order ODEs and systems of two second-order ODEs are presented. The OID of approximate Lie algebra is introduced. It is shown that it is possible to construct a special type of OID, which is used for obtaining the first integral of the system considered. Examples of using the algorithm for cases of complete and incomplete inheritance of a Lie algebra are given.
Keywords:
systems of odes with a small parameter, approximate Lie algebras, invariant representation, operator of invariant differentiation.
Citation:
A. A. Gainetdinova, “Integration of systems of ordinary differential equations with a small parameter which admit approximate Lie algebras”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:2 (2018), 143–160
\Bibitem{Gai18}
\by A.~A.~Gainetdinova
\paper Integration of systems of ordinary differential equations with a small parameter which admit approximate Lie algebras
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2018
\vol 28
\issue 2
\pages 143--160
\mathnet{http://mi.mathnet.ru/vuu627}
\crossref{https://doi.org/10.20537/vm180202}
\elib{https://elibrary.ru/item.asp?id=35258683}
Linking options:
https://www.mathnet.ru/eng/vuu627
https://www.mathnet.ru/eng/vuu/v28/i2/p143
This publication is cited in the following 3 articles:
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
O. A. Narmanov, “Invariantnye resheniya dvumernogo uravneniya teploprovodnosti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 29:1 (2019), 52–60
A. A. Kasatkin, A. A. Gainetdinova, “Symbolic and numerical methods for searching symmetries of ordinary differential equations with a small parameter and reducing its order”, Computer Algebra in Scientific Computing (Casc 2019), Lecture Notes in Computer Science, 11661, eds. M. England, W. Koepf, T. Sadykov, W. Seiler, E. Vorozhtsov, Springer, 2019, 280–299