Abstract
Linear ordinary differential equations whose coefficients are infinite (formal) power series given in a truncated form are considered. Computer algebra procedures (implemented in Maple) for constructing solutions of two forms are suggested. The procedures find the greatest number of series terms occurring in the solutions that can be found for the given truncated series—coefficients.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Notes
The package and the Maple session with examples of use of the procedures described are available at the address http://www. ccas.ru/ca/TruncatedSeries
REFERENCES
Abramov, S.A., Ryabenko, A.A., and Khmelnov, D.E., Linear ordinary differential equations and truncated series, Comput. Math. Math. Phys., 2019, vol. 59, no. 10, pp. 1649—1659.
Abramov, S.A., Ryabenko, A.A., and Khmelnov, D.E., Regular solutions of linear ordinary differential equations and truncated series, Comput. Math. Math. Phys., 2020, vol. 60, no. 1, pp. 4–7.
Barkatou, M. and Pflügel, E., An algorithm computing the regular formal solutions of a system of linear differential equations, J. Symbolic Computation, 1999, vol. 28, pp. 569–587.
Abramov, S., Bronstein, M., and Petkovšek, M., On polynomial solutions of linear operator equations, Proc. of ISSAC’95, 1995, pp. 290–296.
Frobenius, G., Integration der linearen Differentialgleichungen mit veränder Koefficienten, J. für die reine und angewandte Mathematik, 1873, vol. 76, pp. 214–235.
Heffter, L., Einleitung in die Theorie der linearen Differentialgleichungen, Leipzig: Teubner, 1894.
Tournier, E., Solutions formelles d’équations différentielles. Le logiciel de calcul formel DESIR Étude théorique et rкlisation, Thèse d’État (Université de Grenoble), 1987.
Pflügel, E., DESIR-II, RT 154, IMAG Grenoble, 1996.
Abramov, S., Bronstein, M., and Khmelnov, D., On regular and logarithmic solutions of ordinary linear differential systems, Proc. of CASC’05, 2005, pp. 1–12.
Abramov, S.A., Barkatou, M.A., and Pfluegel, E., Higher-order linear differential systems with truncated coefficients, Proc. of CASC’2011, 2011, pp. 10–24.
Abramov, S.A. and Barkatou, M.A., Computable infinite power series in the role of coefficients of linear differential systems, Proc. of CASC’2014, 2014, pp. 1–12.
Abramov, S.A. and Khmelnov, D.E., Regular solutions of linear differential systems with power series coefficients, Program. Comput. Software, 2014, vol. 40, no. 2, pp. 98–105.
Abramov, S.A., Ryabenko, A.A., and Khmelnov, D.E., Procedures for searching local solutions of linear differential systems with infinite power series in the role of coefficients, Program. Comput. Software, 2016, vol. 42, no. 2, pp. 98–105.
Maple online help //http://www.maplesoft.com/support/help/
Funding
This work was supported in part by the Russian Foundation for Basic Research, project no. 19-01-00032-a.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by A. Pesterev
Rights and permissions
About this article
Cite this article
Abramov, S.A., Ryabenko, A.A. & Khmelnov, D.E. Procedures for Searching Laurent and Regular Solutions of Linear Differential Equations with the Coefficients in the Form of Truncated Power Series. Program Comput Soft 46, 67–75 (2020). https://doi.org/10.1134/S0361768820020024
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0361768820020024