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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)

Abstract

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.

Keywords

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

Notes

Acknowledgments

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.

References

  1. 1.
    Olver, P.J.: Applications of Lie Groups to Differential Equations, 1st edn. Springer-Verlag, New York (1986).  https://doi.org/10.1007/978-1-4684-0274-2CrossRefzbMATHGoogle Scholar
  2. 2.
    Ovsyannikov, L.V.: Group Analysis of Differential Equations, 1st edn. Academic Press, New York (1982) Google Scholar
  3. 3.
    Ibragimov, N.H.: Elementary Lie Group Analysis and Ordinary Differential Equations. John Wiley and Sons, Chichester (1999)zbMATHGoogle Scholar
  4. 4.
    Baikov, V.A., Gazizov, R.K., Ibragimov, N.H.: Approximate groups of transformations. Differ. Uravn. 29(10), 1712–1732 (1993). (in Russian)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ayub, M., Mahomed, F.M., Khan, M., Qureshi, M.N.: Symmetries of second-order systems of ODEs and integrability. Nonlinear Dyn. 74, 969–989 (2013).  https://doi.org/10.1007/s11071-013-1016-3MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Wafo Soh, C., Mahomed, F.M.: Reduction of order for systems of ordinary differential equations. J. Nonlinear Math. Phys. 11(1), 13–20 (2004).  https://doi.org/10.2991/jnmp.2004.11.1.3MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gainetdinova, A.A., Gazizov, R.K.: Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras. Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 473(2197), 20160461 (2017).  https://doi.org/10.1098/rspa.2016.0461. 13 ppMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gazizov, R.K., Gainetdinova, A.A.: Operator of invariant differentiation and its application for integrating systems of ordinary differential equations. Ufa Math. J. 9(4), 12–21 (2017).  https://doi.org/10.13108/2017-9-4-12MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gainetdinova, A.A.: Integration of systems of ordinary differential equations with a small parameter which admit approximate Lie algebras. Vestnik Udmurtskogo Universiteta: Matematika, Mekhanika, Komp’yuternye Nauki 28(2), 143–160 (2018).  https://doi.org/10.20537/vm180202. (in Russian)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, The Netherlands (2006)zbMATHGoogle Scholar
  11. 11.
    Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Yu.: Symmetries and group invariant solutions of fractional ordinary differential equations. In: Kochubei, A., Luchko, Yu. (eds.) Fractional Differential Equations, pp. 65–90. De Gruyter, Berlin (2019).  https://doi.org/10.1515/9783110571660zbMATHGoogle Scholar
  12. 12.
    Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Yu.: Symmetries, conservation laws and group invariant solutions of fractional PDEs. In: Kochubei, A., Luchko, Yu. (eds.) Fractional Differential Equations, pp. 353–382. De Gruyter, Berlin (2019).  https://doi.org/10.1515/9783110571660zbMATHGoogle Scholar
  13. 13.
    Galaktionov, V.A., Svirshchevskii, S.R.: Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics. Chapman and Hall/CRC (2006).  https://doi.org/10.1201/9781420011623
  14. 14.
    Gazizov, R.K., Kasatkin, A.A.: Construction of exact solutions for fractional order differential equations by the invariant subspace method. Comput. Math. Appl. 66(5), 576–584 (2013).  https://doi.org/10.1016/j.camwa.2013.05.006MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sahadevan, R., Bakkyaraj, T.: Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations. Fractional Calc. Appl. Anal. 18(1), 146–162 (2015).  https://doi.org/10.1515/fca-2015-0010MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Yu.: Linearly autonomous symmetries of the ordinary fractional differential equations. In: Proceedings of 2014 International Conference on Fractional Differentiation and Its Applications (ICFDA 2014), pp. 1–6. IEEE (2014).  https://doi.org/10.1109/ICFDA.2014.6967419
  17. 17.
    Kasatkin, A.A.: Symmetry properties for systems of two ordinary fractional differential equations. Ufa Math. J. 4(1), 71–81 (2012)MathSciNetGoogle Scholar
  18. 18.
    Hereman, W.: Review of symbolic software for Lie symmetry analysis. Math. Comput. Model. 25(8–9), 115–132 (1997).  https://doi.org/10.1016/S0895-7177(97)00063-0MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cheviakov, A.F.: Symbolic computation of local symmetries of nonlinear and linear partial and ordinary differential equations. Math. Comput. Sci. 4(2–3), 203–222 (2010).  https://doi.org/10.1007/s11786-010-0051-4MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vu, K.T., Jefferson, G.F., Carminati, J.: Finding higher symmetries of differential equations using the MAPLE package DESOLVII. Comput. Phys. Commun. 183(4), 1044–1054 (2012).  https://doi.org/10.1016/j.cpc.2012.01.005MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jefferson, G.F., Carminati, J.: FracSym: automated symbolic computation of Lie symmetries of fractional differential equations. Comput. Phys. Commun. 185(1), 430–441 (2014).  https://doi.org/10.1016/j.cpc.2013.09.019CrossRefzbMATHGoogle Scholar
  22. 22.
    Merkt, B., Timmer, J., Kaschek, D.: Higher-order Lie symmetries in identifiability and predictability analysis of dynamic models. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 92(1), 012920 (2015).  https://doi.org/10.1103/PhysRevE.92.012920MathSciNetCrossRefGoogle Scholar
  23. 23.
    Golub, G.H., Van Loan, C.F.: Matrix Computations. JHU Press, Baltimore (2012)zbMATHGoogle Scholar
  24. 24.
    Bagderina, Y.Y., Gazizov, R.K.: Invariant representation and symmetry reduction for differential equations with a small parameter. Commun. Nonlinear Sci. Num. Simul. 9(1), 3–11 (2004).  https://doi.org/10.1016/S1007-5704(03)00010-8MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Meurer, A., et al.: SymPy: symbolic computing in Python. PeerJ Comput. Sci. 3, e103 (2017).  https://doi.org/10.7717/peerj-cs.103CrossRefGoogle Scholar
  26. 26.
    Gainetdinova, A.A.: Computer program registration certificate 2018618063. Federal Service for Intellectual Property (Rospatent). Registered on 07 September 2018Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Ufa State Aviation Technical UniversityUfaRussia

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