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Lie symmetries for Lie systems: Applications to systems of ODEs and PDEs

Author

Listed:
  • Estévez, P.G.
  • Herranz, F.J.
  • de Lucas, J.
  • Sardón, C.

Abstract

A Lie system is a nonautonomous system of first-order differential equations admitting a superposition rule, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants. Using that a Lie system can be considered as a curve in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra, we associate every Lie system with a Lie algebra of Lie point symmetries induced by the Vessiot–Guldberg Lie algebra. This enables us to derive Lie symmetries of relevant physical systems described by first- and higher-order systems of differential equations by means of Lie systems in an easier way than by standard methods. A generalization of our results to partial differential equations is introduced. Among other applications, Lie symmetries for several new and known generalizations of the real Riccati equation are studied.

Suggested Citation

  • Estévez, P.G. & Herranz, F.J. & de Lucas, J. & Sardón, C., 2016. "Lie symmetries for Lie systems: Applications to systems of ODEs and PDEs," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 435-452.
    • Handle: RePEc:eee:apmaco:v:273:y:2016:i:c:p:435-452
    DOI: 10.1016/j.amc.2015.09.078
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