Abstract
Linearization criteria for two-dimensional systems of second-order ordinary differential equations (ODEs) have been derived earlier using complex symmetry analysis. For such systems, the linearizable form, linearization criteria and symmetry group classification are presented. In this paper, we extend the complex approach to obtain a complex-linearizable form of two-dimensional systems of third-order ODEs. This form leads to a linearizable class and linearization criteria of these systems of ODEs.
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Ali, S.: Complex Symmetry Analysis. PhD Thesis, National University of Sciences and Technology (2008)
Ali, S.; Mahomed, F.M.; Qadir, A.: Linearization criteria for systems of two second order differential equations by complex methods. Nonlinear Dyn. 66, 77 (2011)
Ali, S.; Safdar, M.; Qadir, A.: Linearization from complex Lie point transformations. J. Appl. Math. 2014, 793247 (2014)
Bagderina, Y.Y.: Linearization criteria for a system of two second order ordinary differential equations. J. Phys. A 46, 465201 (2010)
Chern, S.S.: The geometry of the differential equation $y^{\prime \prime \prime \prime }=F(x,y,y^{\prime \prime },y^{\prime \prime \prime })$. Tensor N.S. 28, 173 (1940)
Dutt, H.M.; Safdar, M.: Linearization of two dimensional complex-linearizable systems of second order ordinary differential equations. Appl. Math. Sci. 9, 2889 (2015)
Grebot, G.: The characterization of third order ordinary differential equations admitting a transitive fibre-preserving point symmetry group. J. Math. Anal. Appl. 206, 364 (1997)
Ibragimov, N.H.; Meleshko, S.V.: Linearization of third order ordinary differential equations by point and contact transformations. J. Math. Anal. Appl. 308, 266 (2005)
Ibragimov, N.H.; Meleshko, S.V.; Suksern, S.: Linearization of fourth-order ordinary differential equations by point transformations. J. Phys. A Math. Theor. 41, 235206 (2008)
Lie, S.: Theorie der transformationsgruppen. Math. Ann. 16, 441 (1880)
Lie, S.: Klassifikation und integration von gewönlichen differentialgleichungenzwischen $x, y$, die eine Gruppe von transformationen gestaten. Arch. Math. VIII, IX, 187 (1883)
Mahomed, F.M.; Leach, P.G.L.: Symmetry Lie algebras of $n$th order ordinary differential equations. J. Math. Anal. Appl. 151, 80 (1990)
Mahomed, F.M.: Point symmetry group classification of ordinary differential equation: a survey of some results. Math. Methods Appl. Sci. 30, 1995 (2007)
Mahomed, F.M.; Qadir, A.: Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations. Nonlinear Dyn. 48, 417 (2007)
Mahomed, F.M.; Qadir, A.: Invariant linearization criteria for systems of cubically semi-linear second order ordinary differential equations. J. Nonlinear Math. Phys. 16, 1 (2009)
Neut, S.; Petitot, M.: La géométrie de l’équation $y^{\prime \prime \prime }=f(x, y, y^{\prime }, y^{\prime \prime })$. C. R. Acad. Sci. Paris Sér I(335), 515 (2002)
Sookmee, S.; Meleshko, S.V.: Linearization of two second-order ordinary differential equations via fibre preserving point transformations. ISRN Math. Anal. 2011, 452689 (2001)
Sookmee, S.; Meleshko, S.V.: Conditions for linearization of a projectable system of two second order ordinary differential equations. J. Phys. A 41, 402001 (2008)
Suksern, S.; Meleshko, S.V.; Ibragimov, N.H.: Criteria for fourth order ordinary differential equations to be linearizable by contact transformations. Commun. Nonlinear Sci. Numer. Simul. 14, 2619 (2009)
Tressé, A.: Sur les Invariants differentiels des groupes continus de transformations. Acta. Math. 18, 1 (1894)
Wafo Soh, C.; Mahomed, F.M.: Linearization criteria for a system of second order ordinary differential equations. Int. J. Nonlinear Mech. 36, 671 (2001)
Acknowledgements
Two of us (HMD and AQ) are grateful to the Higher Education Commission of Pakistan for support under their Project No. 3054.
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Dutt, H.M., Safdar, M. & Qadir, A. Linearization criteria for two-dimensional systems of third-order ordinary differential equations by complex approach. Arab. J. Math. 8, 163–170 (2019). https://doi.org/10.1007/s40065-019-0238-8
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DOI: https://doi.org/10.1007/s40065-019-0238-8