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The Lie Algebra of Vector Fields Package with Applications to Mappings of Differential Equations

  • Zahra MohammadiEmail author
  • Gregory J. Reid
  • S.-L. Tracy Huang
Conference paper
  • 44 Downloads
Part of the Communications in Computer and Information Science book series (CCIS, volume 1125)

Abstract

Lie symmetry groups of transformations (mappings) of differential equations leave them invariant, and are most conveniently studied through their Lie algebra of vector fields (essentially the linearization of the mappings around the identity transformation). Maple makes powerful and frequent use of such Lie algebras, mostly through routines that are dependent on Maple’s powerful exact integration routines, that essentially automate traditional hand-calculation strategies. However these routines are usually heuristic, and algorithmic approaches require a deeper integration of differential elimination (differential algebraic) approaches in applications to differential equations. This is the underlying motivation of the LieAlgebrasOfVectorFields (LAVF) package of Huang and Lisle. The LAVF package introduces a powerful algorithmic calculus for doing calculations with differential equations without the heuristics of integration to calculate efficiently many properties of such systems.

We use LAVF in the development of our MapDE package, which determines the existence of analytic invertible mappings of an input DE to target DE. Theory, algorithms, and examples of MapDE can be found in [5, 6]. Here we present a brief summary, through examples, of the application of LAVF to MapDE.

Keywords

Symmetry Lie algebra Structure constants Differential algebra 

References

  1. 1.
    Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-0-387-68028-6 CrossRefzbMATHGoogle Scholar
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    Huang, S.L.: Properties of lie algebras of vector fields from lie determining system. Ph.D. thesis, University of Canberra (2015)Google Scholar
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    Lisle, I.G., Huang, S.-L.T: Algorithmic calculus for Lie determining systems. J. Symbolic Comput. 79(part 2), 482–498 (2017)MathSciNetCrossRefGoogle Scholar
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    Lyakhov, D., Gerdt, V., Michels, D.: Algorithmic verification of linearizability for ordinary differential equations. In: Proceedings ISSAC 2017, pp. 285–292. ACM (2017)Google Scholar
  5. 5.
    Mohammadi, Z., Reid, G., Huang, S.-L.T.: Introduction of the MapDE algorithm for determination of mappings relating differential equations. arXiv:1903.02180v1 [math.AP] (2019). To appear in Proceedings of ISSAC 2019. ACM
  6. 6.
    Mohammadi, Z., Reid, G., Huang, S.-L.T.: Symmetry-based algorithms for invertible mappings of polynomially nonlinear PDE to linear PDE. Submitted to Mathematics of Computer Science (Revision requested 15 May 2019)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Zahra Mohammadi
    • 1
    Email author
  • Gregory J. Reid
    • 1
  • S.-L. Tracy Huang
    • 2
  1. 1.Department of Applied MathematicsUniversity of Western OntarioLondonCanada
  2. 2.Data61, CSIROCanberraAustralia

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