An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator that generates an iterative equation of a general order in reduced normal form is also obtained and some other properties of iterative equations are established. An expression for the parameters of the source equation of the transformed equation under equivalence transformations is obtained, and this gives rise to the derivation of important symmetry properties for iterative equations. The transformation mapping a given iterative equation to the canonical form is obtained in terms of the simplest determining equation, and several examples of application are discussed.
@article{bwmeta1.element.doi-10_2478_s11533-013-0364-z,
author = {Jean-Claude Ndogmo and Fazal Mahomed},
title = {On certain properties of linear iterative equations},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {648-657},
zbl = {1314.34032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0364-z}
}
Jean-Claude Ndogmo; Fazal Mahomed. On certain properties of linear iterative equations. Open Mathematics, Tome 12 (2014) pp. 648-657. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0364-z/
[1] Cariñena J.F., Grabowski J., de Lucas J., Superposition rules for higher-order systems and their applications, J. Phys. A, 2012, 45(18), #185202 http://dx.doi.org/10.1088/1751-8113/45/18/185202 | Zbl 1248.34008
[2] Krause J., Michel L., Equations différentielles linéaires d’ordre n > 2 ayant une algèbre de Lie de symétrie de dimension n + 4, C. R. Acad. Sci. Paris, 1988, 307(18), 905–910 | Zbl 0662.34010
[3] Krause J., Michel L., Classification of the symmetries of ordinary differential equations, In: Group Theoretical Methods in Physics, Moscow, June 4–9, 1990, Lecture Notes in Phys., 382, Springer, Berlin, 1991, 251–262
[4] Leach P.G.L., Andriopoulos K., The Ermakov equation: a commentary, Appl. Anal. Discrete Math., 2008, 2(2), 146–157 http://dx.doi.org/10.2298/AADM0802146L | Zbl 1199.34006
[5] Lie S., Classification und Integration von gewöhnlichen Differentialgleichungen zwischen xy, die eine Gruppe von Transformationen gestatten. III, Archiv for Mathematik og Naturvidenskab, 1883, 8, 371–458 | Zbl 15.0751.03
[6] Lie S., Classification und Integration von gewöhnlichen Differentialgleichungen zwischen xy, die eine Gruppe von Transformationen gestetten, Math. Ann., 1888, 32(2), 213–281 http://dx.doi.org/10.1007/BF01444068
[7] de Lucas J., Sardón C., On Lie systems and Kummer-Schwarz equations, J. Math. Phys., 2013, 54(3), #033505 http://dx.doi.org/10.1063/1.4794280 | Zbl 1312.34032
[8] Mahomed F.M., Leach P.G.L., Symmetry Lie algebras of nth order ordinary differential equations, J. Math. Anal. Appl., 1990, 151(1), 80–107 http://dx.doi.org/10.1016/0022-247X(90)90244-A | Zbl 0719.34018
[9] Ndogmo J.C., Equivalence transformations of the Euler-Bernoulli equation, Nonlinear Anal. Real World Appl., 2012, 13(5), 2172–2177 http://dx.doi.org/10.1016/j.nonrwa.2012.01.012 | Zbl 1257.35006
[10] Ndogmo J.C., Some results on equivalence groups, J. Appl. Math., 2012, #484805 | Zbl 1280.34041
[11] Schwarz F., Solving second order ordinary differential equations with maximal symmetry group, Computing, 1999, 62(1), 1–10 http://dx.doi.org/10.1007/s006070050009 | Zbl 0934.34001
[12] Schwarz F., Equivalence classes, symmetries and solutions of linear third-order differential equations, Computing, 2002, 69(2), 141–162 http://dx.doi.org/10.1007/s00607-002-1454-0 | Zbl 1025.34005
[13] Sebbar A., Sebbar A., Eisenstein series and modular differential equations, Canad. Math. Bull., 2012, 55(2), 400–409 http://dx.doi.org/10.4153/CMB-2011-091-3 | Zbl 1272.11055
[14] Tsitskishvili A., Solution of the Schwarz differential equation, Mem. Differential Equations Math. Phys., 1997, 11, 129–156 | Zbl 0911.34004