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Journal of Applied Analysis

Editor-in-Chief: Fechner, Włodzimierz / Ciesielski, Krzysztof

Managing Editor: Gajek, Leslaw

CiteScore 2018: 0.45

SCImago Journal Rank (SJR) 2018: 0.181
Source Normalized Impact per Paper (SNIP) 2018: 0.845

Mathematical Citation Quotient (MCQ) 2018: 0.20

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1869-6082
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Volume 24, Issue 1

Some variational principles associated with ODEs of maximal symmetry. Part 1: Equations in canonical form

Jean-Claude Ndogmo
Published Online: 2018-05-03 | DOI: https://doi.org/10.1515/jaa-2018-0002

Abstract

Variational and divergence symmetries are studied in this paper for linear equations of maximal symmetry in canonical form, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. Some of these results apply to linear equations of a general form and of arbitrary orders or having a symmetry algebra of arbitrary dimension.

MSC 2010: 34A26; 35A24; 70S10; 37K05

References

• [1]

S. Anco, G. Bluman and T. Wolf, Invertible mappings of nonlinear PDEs to linear PDEs through admitted conservation laws, Acta Appl. Math. 101 (2008), no. 1–3, 21–38.

• [2]

J. J. H. Bashingwa, A. H. Bokhari, A. H. Kara and F. D. Zaman, The geometry and invariance properties for certain classes of metrics with neutral signature, Int. J. Geom. Methods Mod. Phys. 13 (2016), no. 6, Article ID 1650080.

• [3]

G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Appl. Math. Sci. 81, Springer, New York, 1989. Google Scholar

• [4]

R. de la Rosa, M. L. Gandarias and M. S. Bruzón, On symmetries and conservation laws of a Gardner equation involving arbitrary functions, Appl. Math. Comput. 290 (2016), 125–134.

• [5]

G. P. Flessas, K. S. Govinder and P. G. L. Leach, Characterisation of the algebraic properties of first integrals of scalar ordinary differential equations of maximal symmetry, J. Math. Anal. Appl. 212 (1997), no. 2, 349–374.

• [6]

J. Krause and L. Michel, Équations différentielles linéaires d’ordre $n\ge 2$ ayant une algèbre de Lie de symétrie de dimension $n+4$, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 18, 905–910. Google Scholar

• [7]

S. Lie, Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen xy, die eine Gruppe von Transformationen gestatten, Math. Ann. 32 (1888), no. 2, 213–281.

• [8]

A. B. Magan, D. P. Mason and F. M. Mahomed, Analytical solution in parametric form for the two-dimensional free jet of a power-law fluid, Int. J. Nonlinear Mech. 851 (2016), 94–108. Google Scholar

• [9]

F. M. Mahomed and P. G. L. Leach, Symmetry Lie algebras of nth order ordinary differential equations, J. Math. Anal. Appl. 151 (1990), no. 1, 80–107.

• [10]

J. C. Ndogmo, Generation and identification of ordinary differential equations of maximal symmetry algebra, Abstr. Appl. Anal. 2016 (2016), Article ID 1796316. Google Scholar

• [11]

J. C. Ndogmo, Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case, J. Appl. Anal. 24 (2018), to appear. Google Scholar

• [12]

J.-C. Ndogmo and F. M. Mahomed, On certain properties of linear iterative equations, Cent. Eur. J. Math. 12 (2014), no. 4, 648–657.

• [13]

P. J. Olver, Applications of Lie Groups to Differential Equations, Grad. Texts in Math. 107, Springer, New York, 1986. Google Scholar

Accepted: 2017-12-19

Published Online: 2018-05-03

Published in Print: 2018-06-01

This research is supported by research grants from the University of Venda (grant number I538) and from the NRF of South Africa (grant number 97822).

Citation Information: Journal of Applied Analysis, Volume 24, Issue 1, Pages 17–26, ISSN (Online) 1869-6082, ISSN (Print) 1425-6908,

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