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Abstract and Applied Analysis
Volume 2016, Article ID 1796316, 9 pages
http://dx.doi.org/10.1155/2016/1796316
Research Article

Generation and Identification of Ordinary Differential Equations of Maximal Symmetry Algebra

Department of Mathematics and Applied Mathematics, University of Venda, P/B X5050, Thohoyandou, Limpopo 0950, South Africa

Received 13 June 2016; Revised 25 October 2016; Accepted 7 November 2016

Academic Editor: Jaume Giné

Copyright © 2016 J. C. Ndogmo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Krause and L. Michel, “Équations différentielles linéaires d'ordre n>2 ayant une algèbre de Lie de symétrie de dimension n+4,” Comptes Rendus de l'Académie des Sciences, vol. 307, no. 18, pp. 905–910, 1988. View at Google Scholar
  2. F. M. Mahomed and P. G. L. Leach, “Symmetry Lie algebras of nth order ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 151, no. 1, pp. 80–107, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. J.-C. Ndogmo and F. M. Mahomed, “On certain properties of linear iterative equations,” Central European Journal of Mathematics, vol. 12, no. 4, pp. 648–657, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. R. Campoamor-Stursberg, “An alternative approach to systems of second-order ordinary differential equations with maximal symmetry. Realizations sln+2, R by special functions,” Communications in Nonlinear Science and Numerical Simulation, vol. 37, pp. 200–211, 2016. View at Publisher · View at Google Scholar
  5. R. Campoamor-Stursberg and J. Guerón, “Linearizing systems of second-order ODEs via symmetry generators spanning a simple subalgebra,” Acta Applicandae Mathematicae, vol. 127, pp. 105–115, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. K. S. Mahomed and E. Momoniat, “Algebraic properties of first integrals for scalar linear third-order ODEs of maximal symmetry,” Abstract and Applied Analysis, vol. 2013, Article ID 530365, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  7. K. S. Mahomed and E. Momoniat, “Symmetry classification of first integrals for scalar linearizable second-order ODEs,” Journal of Applied Mathematics, vol. 2012, Article ID 847086, 14 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  8. F. Schwarz, “Solving second order ordinary differential equations with maximal symmetry group,” Computing, vol. 62, no. 1, pp. 1–10, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  9. F. Schwarz, “Equivalence classes, symmetries and solutions of linear third-order differential equations,” Computing, vol. 69, no. 2, pp. 141–162, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. P. G. Leach, R. R. Warne, N. Caister, V. Naicker, and N. Euler, “Symmetries, integrals and solutions of ordinary differential equations of maximal symmetry,” Indian Academy of Sciences. Proceedings. Mathematical Sciences, vol. 120, no. 1, pp. 113–130, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. V. P. Ermakov, “Second-order differential equations: conditions of complete integrability,” Applicable Analysis and Discrete Mathematics, vol. 2, no. 2, pp. 123–145, 2008, Translated from the 1880 Russian original by A. O. Harin, and edited by P. G. L. Leach. View at Publisher · View at Google Scholar · View at MathSciNet
  12. J. C. Ndogmo, “Equivalence transformations of the Euler-Bernoulli equation,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2172–2177, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. J. C. Ndogmo, “Some results on equivalence groups,” Journal of Applied Mathematics, vol. 2012, Article ID 484805, 11 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  14. S. Lie, “Klassification und Integration von gewöhnlichen Differentialgleichungen zwischen x; y; die eine Gruppe von Transformationen gestetten. I,” Mathematische Annalen, vol. 22, pp. 213–253, 1888. View at Google Scholar
  15. P. J. Olver, “Differential invariants and invariant differential equations,” Lie Groups and Their Applications, vol. 1, no. 1, pp. 177–192, 1994. View at Google Scholar · View at MathSciNet
  16. P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, UK, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  17. M. Bronstein, T. Mulders, and J.-A. Weil, “On symmetric powers of differential operators,” in Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC '97), pp. 156–163, ACM, 1997. View at Publisher · View at Google Scholar
  18. G. W. Hill, “On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon,” Acta Mathematica, vol. 8, no. 1, pp. 1–36, 1886. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, Providence, RI, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  20. W. Magnus and S. Winkler, Hill's Equation, John Wiley & Sons, New York, NY, USA, 1966.
  21. A. Loewy, “Über vollständig reduzible lineare homogene Differentialgleichungen,” Mathematische Annalen, vol. 62, no. 1, pp. 89–117, 1906. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. F. Schwarz, “Loewy decomposition of linear differential equations,” Bulletin of Mathematical Sciences, vol. 3, no. 1, pp. 19–71, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus