Criteria for System of Three Second-Order Ordinary Differential Equations to Be Reduced to a Linear System via Restricted Class of Point Transformation

Supaporn Suksern, Nawee Sakdadech

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, Thailand.

DOI: 10.4236/am.2014.53053                

Abstract

This paper is devoted to the study of the linearization problem of system of three second-order ordinary differential equations and

. The necessary conditions for linearization by general point transformation and are found. The sufficient conditions for linearization by restricted class of point transformation

and are obtained. Moreover, the procedure for obtaining the linearizing transformation is provided in explicit forms. Examples demonstrating the procedure of using the linearization theorems are presented.

Keywords

Linearization Problem; Point Transformation; System of Three Second-Order Ordinary Differential Equation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. Lie, “Klassifikation und Integration Vongewohnlichen Differentialgleichungenzwischen x, y die eine Gruppe von Transformation engestatten. III,” Archiv for Matematikog Naturvidenskab, Vol. 8, No. 4, 1883, pp. 371-427.
[2]	A. M. Tresse, “Détermination des Invariants Ponctuels de l’équationdifférentielleordinaire du Second Ordre ,” Preisschriften der Fürstlichen Jablonowski'schen Gesellschaft XXXII, Leipzig, 1896.
[3]	R. Liouville, “Sur les Invariants de Certaines Equations Differentielles et Surleurs Applications,” Journal de l’école Polytechnique, Vol. 59, 1889, pp. 7-76.
[4]	N. H. Ibragimov, “Invariants of a Remarkable Family of Nonlinear Equations,” Nonlinear Dynamics, Vol. 30, No. 2, 2002, pp. 155-166. http://dx.doi.org/10.1023/A:1020406015011
[5]	E. Cartan, “Sur les Variétés àconnexion Projective,” Bulletin de la Société Mathématique de France, Vol. 52, 1924, pp. 205-241.
[6]	C. W. Soh and F. M. Mahomed, “Linearization Criteria for a System of Second-Order Ordinary Differential Equations,” International Journal of Non-Linear Mechanics, Vol. 36, No. 36, 2001, pp. 671-677. http://dx.doi.org/10.1016/S0020-7462(00)00032-9
[7]	S. Sookmee and S. V. Meleshko, “Linearization of Two Second-Order Ordinary Differential Equations via Fiber Preserving Point Transformations,” ISRN Mathematical Analysis, Vol. 2001, 2011, Article ID: 452689. http://dx.doi.org/10.5402/2011/452689
[8]	A. V. Aminova and N. A.-M. Aminov, “Projective Geometry of Systems of Second-Order Differential Equations,” Sbornik: Mathematics, Vol. 197, No. 7, 2006, pp. 951-955. http://dx.doi.org/10.1070/SM2006v197n07ABEH003784
[9]	F. M. Mahomed and A. Qadir, “Linearization Criteria for a System of Second-Order Quadratically Semi-Linear Ordinary Differential Equations,” Nonlinear Dynamics, Vol. 48, No. 4, 2007, pp. 417-422. http://dx.doi.org/10.1007/s11071-006-9095-z
[10]	S. Sookmee, “Invariants of the Equivalence Group of a System of Second-Order Ordinary Differential Equations,” M.S. Thesis, 2005.
[11]	S. Sookmee and S. V. Meleshko, “Conditions for Linearization of a Projectable System of Two Second-Order Ordinary Differential Equations,” Journal of Physics A: Mathematical and Theoretical, Vol. 41, No. 402001, 2008, pp. 1-7.
[12]	S. Neut, M. Petitot and R. Dridi, “élieCartan’s Geometrical Vision or How to Avoid Expression Swell,” Journal of Symbolic Computation, Vol. 44, No. 3, 2009, pp. 261-270. http://dx.doi.org/10.1016/j.jsc.2007.04.006
[13]	Y. Y. Bagderina, “Linearization Criteria for a System of Two Second-Order Ordinary Differential Equations,” Journal of Physics A, Vol. 43, No. 46, 2010, Article ID: 465201.