Abstract
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.
Similar content being viewed by others
References
Cariñena J.F., Grabowski J., de Lucas J., Superposition rules for higher-order systems and their applications, J. Phys. A, 2012, 45(18), #185202
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
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
Leach P.G.L., Andriopoulos K., The Ermakov equation: a commentary, Appl. Anal. Discrete Math., 2008, 2(2), 146–157
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
Lie S., Classification und Integration von gewöhnlichen Differentialgleichungen zwischen xy, die eine Gruppe von Transformationen gestetten, Math. Ann., 1888, 32(2), 213–281
de Lucas J., Sardón C., On Lie systems and Kummer-Schwarz equations, J. Math. Phys., 2013, 54(3), #033505
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
Ndogmo J.C., Equivalence transformations of the Euler-Bernoulli equation, Nonlinear Anal. Real World Appl., 2012, 13(5), 2172–2177
Ndogmo J.C., Some results on equivalence groups, J. Appl. Math., 2012, #484805
Schwarz F., Solving second order ordinary differential equations with maximal symmetry group, Computing, 1999, 62(1), 1–10
Schwarz F., Equivalence classes, symmetries and solutions of linear third-order differential equations, Computing, 2002, 69(2), 141–162
Sebbar A., Sebbar A., Eisenstein series and modular differential equations, Canad. Math. Bull., 2012, 55(2), 400–409
Tsitskishvili A., Solution of the Schwarz differential equation, Mem. Differential Equations Math. Phys., 1997, 11, 129–156
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Ndogmo, JC., Mahomed, F.M. On certain properties of linear iterative equations. centr.eur.j.math. 12, 648–657 (2014). https://doi.org/10.2478/s11533-013-0364-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11533-013-0364-z