Symbolic and Numerical Methods for Searching Symmetries of Ordinary Differential Equations with a Small Parameter and Reducing Its Order

  • Alexey A. KasatkinEmail author
  • Aliya A. Gainetdinova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)


Two programs for computer algebra systems are described that deal with Lie algebras of generators admitted by systems of ordinary differential equations (ODEs). The first one allows to find the generators of admitted transformations in the specified form. This program is written in Python and based on SciPy library. It does not require solving partial differential equations symbolically and can also analyze equations with Riemann–Liouville fractional derivatives and find approximate symmetries for systems of equations with a small parameter. The second program written as a package for Maple computes the operator of invariant differentiation in special form for given Lie algebra of generators. This operator is used for order reduction of given ODE systems.


Lie algebras of generators Point symmetries of differential equation Fractional derivatives Differential invariants Operators of invariant differentiation Computer algebra 



We are grateful to Prof. R.K. Gazizov and Prof. S.Yu. Lukashchuk for constructive discussion. Also we thank the referees whose comments helped us a lot to improve the early draft of this paper.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Ufa State Aviation Technical UniversityUfaRussia

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