Skip to main content
SpringerLink
Log in

On Implementation of Numerical Methods for Solving Ordinary Differential Equations in Computer Algebra Systems

  • Published:
Programming and Computer Software Aims and scope Submit manuscript

Abstract

This paper presents an original package for investigating numerical solutions of ordinary differential equations, which is built in the Sage computer algebra system. This project is focused on a closer integration of numerical and symbolic methods while primarily aiming to create a convenient tool for working with numerical solutions in Sage. The package defines two new classes: initial problems and approximate solutions. The first class defines tools for symbolic computations related to initial problems, while the second class defines tools for interpolating values of symbolic expressions on an approximate solution and estimating the error with the use of the Richardson method. An implementation of the Runge–Kutta method is briefly described, with its main feature being the possibility of working with arbitrary Butcher tableaux and arbitrary numeric fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.

Similar content being viewed by others

REFERENCES

  1. Runge, C. and König, H., Vorlesungen über numerisches Rechnen, Springer-Verlag, 2013.

  2. SciPy documentation, 2022. https://docs.scipy.org.

  3. Ketcheson, D.I. and Waheed, U., A comparison of high order explicit Runge–Kutta, extrapolation, and deferred correction methods in serial and parallel, CAMCoS, 2014, vol. 9, no. 2, pp. 175–200.

    Article  MathSciNet  MATH  Google Scholar 

  4. Gevorkyan, M.N., Concrete implementations of symplectic numerical methods, Vestn. Ross. Univ. Druzhby Nar., Ser. Mat., Inf., Fiz., 2013, no. 3, pp. 77–89.

  5. Castillo, J.E. and Miranda, G.F., Mimetic Discretization Methods, Chapman and Hall/CRC, 2013.

    Book  Google Scholar 

  6. Da Veiga, L.B., Lipnikov, K., and Manzini, G., The mimetic finite difference method for elliptic problems, 2014, vol. 11.

    Book  MATH  Google Scholar 

  7. Hairer, E., Wanner, G., and Lubich, Ch., Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, New York: Springer, 2000.

    MATH  Google Scholar 

  8. Ying, Yu, Baddour, A., Gerdt, V.P., et al., On the quadratization of the integrals for the many-body problem, Mathematics, 2021, vol. 9, no. 24.

  9. Baddour, A. and Malykh, M., On difference schemes for the many-body problem preserving all algebraic integrals, Phys. Part. Nucl. Lett., 2022, vol. 19, pp. 77–80.

    Article  Google Scholar 

  10. Baddour, A., Malykh, M., and Sevastianov, L., On periodic approximate solutions of dynamical systems with quadratic right-hand side, J. Math. Sci., 2022, vol. 261, pp. 698–708.

    Article  MathSciNet  MATH  Google Scholar 

  11. Stein, W.A., Sage Mathematics Software (Version 6.7), 2015. http:// www.sagemath.org.

  12. Malykh, M.D. and Ying, Yu, Technique for finding algebraic integrals of first-order differential equations, Vestn. Ross. Univ. Druzhby Nar., Ser. Mat., Inf., Fiz., 2018, vol. 26, no. 3, pp. 285–291.

    Google Scholar 

  13. Kalitkin, N.N., Al’shin, A.B., Al’shina, E.A., and Rogov, B.V., Vychisleniya na kvaziravnomernykh setkakh (Computations on Quasi-Uniform Grids), Moscow: Fizmatlit, 2005.

  14. Belov, A.A., Kalitkin, N.N., and Poshivaylo, I.P., Geometrically adaptive grids for stiff Cauchy problems, Dokl. Math., 2016, vol. 93, no. 1, pp. 112–116.

    Article  MathSciNet  MATH  Google Scholar 

  15. Belov, A.A. and Kalitkin, N.N., Nonlinearity problem in the numerical solution of superstiff Cauchy problems, Math. Models Comput. Simul., 2016, vol. 8, no. 6, pp. 638–650.

    Article  MathSciNet  Google Scholar 

  16. Belov, A.A., Kalitkin, N.N., Bulatov, P.E., and Zholkovskii, E.K., Explicit methods for integrating stiff Cauchy problems, Dokl. Math., 2019, vol. 99, no. 2, pp. 230–234.

    Article  MathSciNet  MATH  Google Scholar 

  17. Baddour, A. and Malykh, M.D., Richardson–Kalitkin method in abstract description, Discrete Contin. Models Appl. Comput. Sci., 2021, vol. 29, no. 3, pp. 271–284.

    Article  Google Scholar 

  18. Baddour, A., Malykh, M.D., Panin, A.A., and Sevastianov, L.A., Numerical determination of the singularity order of a system of differential equations, Discrete Contin. Models Appl. Comput. Sci., 2020, vol. 28, no. 1, pp. 17–34.

    Article  Google Scholar 

  19. Hairer, E., Wanner, G., and Norsett, S.P., Solving Ordinary Differential Equations I, Springer, 2008, 3rd ed.

    MATH  Google Scholar 

  20. Ying, Yu, The symbolic problems associated with Runge–Kutta methods and their solving in Sage, Discrete Contin. Models Appl. Comput. Sci., 2019, vol. 27, no. 1, pp. 33–41.

    Google Scholar 

  21. Khashin, S.I., Numerical solution of Butcher equations, Vestn. Ivanovskogo Gos. Univ., 2000, no. 3, pp. 155–164.

  22. Khammud, G.M. and Khashin, S.I., Six-dimensional family of 5-order 6-step Runge–Kutta methods, Nauch. Tr. Ivanovskogo Gos. Univ., Mat., 2001, no. 4, pp. 114–122.

  23. Khashin, S.I., Alternative form of Butcher equations, Vestn. Ivanovskogo Gos. Univ., 2007, no. 3, pp. 94–103.

  24. Khashin, S.I., A symbolic-numeric approach to the solution of the Butcher equations, Can. Appl. Math. Q., 2009, vol. 17, no. 3, pp. 555–569.

    MathSciNet  MATH  Google Scholar 

  25. Khashin, S.I., Three simplifying assumptions for Runge–Kutta Methods, Vestn. Ivanovskogo Gos. Univ., 2012, no. 2, pp. 142–150.

  26. Stone, P., Maple worksheets on the derivation of Runge–Kutta schemes, 2021. http://www.peterstone.name/Maplepgs/RKcoeff.html.

  27. Sikorskii, Yu.S., Elementy teorii ellipticheskikh funktsii s prilozheniyami k mekhanike (Elements of the Theory of Elliptic Functions with Applications to Mechanics), Moscow: ONTI, 1936.

  28. Scarborough, J.B., Numerical Mathematical Analysis, Oxford University Press, 1930.

    MATH  Google Scholar 

Download references

Funding

This work was supported by the Russian Science Foundation, project no. 20-11-20257.

Author information

Authors and Affiliations

  1. Peoples’ Friendship University of Russia, ul. Miklukho-Maklaya 6, 117198, Moscow, Russia

    A. Baddour, M. M. Gambaryan, L. Gonzalez & M. D. Malykh

  2. Joint Institute for Nuclear Research, 141980, Dubna, Moscow oblast, Russia

    M. D. Malykh

Authors
  1. A. Baddour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. M. Gambaryan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Gonzalez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. D. Malykh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. D. Malykh.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

Translated by Yu. Kornienko

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baddour, A., Gambaryan, M.M., Gonzalez, L. et al. On Implementation of Numerical Methods for Solving Ordinary Differential Equations in Computer Algebra Systems. Program Comput Soft 49, 412–422 (2023). https://doi.org/10.1134/S0361768823020044

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768823020044

Search

Navigation