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Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability
Joint Authors
Ayub, Muhammad
Khan, Masood
Mahomed, Fazal Mahmood
Source
Journal of Applied Mathematics
Issue
Vol. 2013, Issue 2013 (31 Dec. 2013), pp.1-15, 15 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2013-04-03
Country of Publication
Egypt
No. of Pages
15
Main Subjects
Abstract EN
We present a systematic procedure for the determination of a complete set of kth-order (k≥2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras.
In addition, we give a procedure for the construction of a system of two kth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that there are 29 classes for the case of k = 2 and 31 classes for the case of k≥3.
We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras.
Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations.
We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations.
A natural extension of this result is provided for a system of two kth-order (k≥3) ODEs.
We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.
American Psychological Association (APA)
Ayub, Muhammad& Khan, Masood& Mahomed, Fazal Mahmood. 2013. Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability. Journal of Applied Mathematics،Vol. 2013, no. 2013, pp.1-15.
https://search.emarefa.net/detail/BIM-449554
Modern Language Association (MLA)
Ayub, Muhammad…[et al.]. Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability. Journal of Applied Mathematics No. 2013 (2013), pp.1-15.
https://search.emarefa.net/detail/BIM-449554
American Medical Association (AMA)
Ayub, Muhammad& Khan, Masood& Mahomed, Fazal Mahmood. Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability. Journal of Applied Mathematics. 2013. Vol. 2013, no. 2013, pp.1-15.
https://search.emarefa.net/detail/BIM-449554
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-449554