Journal of Nonlinear Mathematical Physics

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Volume 16, Issue 3, September 2009, Pages 283 - 298

Invariant Linearization Criteria for Systems of Cubically Nonlinear Second-Order Ordinary Differential Equations

Authors
F. M. Mahomed
School of Computational and Applied Mathematics, Centre for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Wits 2050, South Africa,Fazal.Mahomed@wits.ac.za
Asghar Qadir
Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus of the College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistan
Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia,aqadirs@comsats.net.pk
Received 14 November 2008, Accepted 18 February 2009, Available Online 7 January 2021.
DOI
10.1142/S1402925109000236How to use a DOI?
Keywords
Invariant criteria; second-order systems; linearization; geometric approach; Lie algebras
Abstract

Invariant linearization criteria for square systems of second-order quadratically nonlinear ordinary differential equations (ODEs) that can be represented as geodesic equations are extended to square systems of ODEs cubically nonlinear in the first derivatives. It is shown that there are two branches for the linearization problem via point transformations for an arbitrary system of second-order ODEs and its reduction to the simplest system. One is when the system is at most cubic in the first derivatives. One obtains the equivalent of the Lie conditions for such systems. We explicitly solve this branch of the linearization problem by point transformations in the case of a square system of two second-order ODEs. Necessary and sufficient conditions for linearization to the simplest system by means of point transformations are given in terms of coefficient functions of the system of two second-order ODEs cubically nonlinear in the first derivatives. A consequence of our geometric approach of projection is a rederivation of Lie’s linearization conditions for a single second-order ODE and sheds light on more recent results for them. In particular we show here how one can construct point transformations for reduction to the simplest linear equation by going to the higher space and just utilizing the coefficients of the original ODE. We also obtain invariant criteria for the reduction of a linear square system to the simplest system. Moreover these results contain the quadratic case as a special case. Examples are given to illustrate our results.

Copyright
© 2009 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
16 - 3
Pages
283 - 298
Publication Date
2021/01/07
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1142/S1402925109000236How to use a DOI?
Copyright
© 2009 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

risenwbib
TY  - JOUR
AU  - F. M. Mahomed
AU  - Asghar Qadir
PY  - 2021
DA  - 2021/01/07
TI  - Invariant Linearization Criteria for Systems of Cubically Nonlinear Second-Order Ordinary Differential Equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 283
EP  - 298
VL  - 16
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925109000236
DO  - 10.1142/S1402925109000236
ID  - Mahomed2021
ER  -