Procedures to search for Laurent and regular solutions of linear ordinary differential equations with truncated power series coefficients

We previously published algorithms for searching the so-called Laurent and regular solutions of linear ordinary differential equations with infinite formal power series in the role of coefficients. The question of infinite series representation is very important for computer algebra. In those algorithms the series are given in truncated form, which means that we do not have complete information about the equation under consideration. Based on this incomplete information, algorithms give the maximum possible number of terms of the series included in the solutions. We are interested in the information about these solutions that is invariant to possible prolongations of those truncated series that represent the coefficients of the equation. The mentioned publications reported preliminary (trial) versions for procedures, which implement these algorithms, as well as experiments with them. To date, the procedures have been improved, the interface and data presentation are designed for them in a uniform manner. The advanced procedures are discussed in the current paper. The various examples are presented which illustrates the use of the procedures, including their optional parameters. These procedures are available from the web page http://www.ccas.ru/ca/TruncatedSeries.

1. Абрамов С.А., Рябенко А.А., Хмельнов Д.Е. Линейные обыкновенные дифференциальные уравнения и усеченные ряды. Журнал вычислительной математики и математической физики, том 59, № 10, 2019, стр. 66–-77 / Abramov S.A., Ryabenko A.A., Khmelnov D.E. Linear ordinary differential equations and truncated series. Computational Mathematics and Mathematical Physics, vol. 59, № 10, 2019.

2. Абрамов С.А., Рябенко А.А., Хмельнов Д.Е. Регулярные решения линейных обыкновенных дифференциальных уравнений и усеченные ряды. Журнал вычислительной математики и математической физики, том 60, № 1, 2020 / Abramov S.A., Ryabenko A.A., Khmelnov D.E. Regular solutions of linear ordinary differential equations and truncated series. Computational Mathematics and Mathematical Physics, vol. 60, № 1, 2020.

3. Moulay Barkatou, Eckhard Pflügel. An algorithm computing the regular formal solutions of a system of linear differential equations. Journal of Symbolic Computation. vol. 28, issues 4–5, 1999, pp. 569-587.

4. Sergei A. Abramov, Manuel Bronstein, Marko Petkovšek. On polynomial solutions of linear operator equations. In Proc. of the 1995 international symposium on Symbolic and algebraic computation. 1995, pp. 290-296.

5. Ferdinand Georg Frobenius. Integration der linearen Differentialgleichungen mit verânder Koefficienten. Journal für die reine und angewandte Mathematik, vol.76, 1873, pp. 214-235.

6. Lothar Heffter. Einleitung in Die Theorie Der Linearen Differentialgleichungen Mit Einer Unabhängigen Variablen. Teubner, Leipzig, 1894, 283 p.

7. Évelyne Tournier. Solutions formelles d'équations différentielles. Le logiciel de calcul formel DESIR, Étude théorique et realization. Thèse d'État, Université I de Grenoble, 1987.

8. Eckhard Pflügel. DESIR-II. RT 154, IMAG Grenoble. 1996.

9. Abramov S., Bronstein M., Khmelnov D. On regular and logarithmic solutions of ordinary linear differential systems. Lecture Notes in Computer Science, vol. 3718, 2005, pp. 1-12.

10. Abramov S.A., Barkatou M.A., Pfluegel E. Higher-order linear differential systems with truncated coefficients. Lecture Notes in Computer Science, vol. 6885, 2011, pp. 10-24.

11. Abramov S.A., Barkatou M.A. Computable Infinite Power Series in the Role of Coefficients of Linear Differential Systems. Lecture Notes in Computer Science, vol. 8660, 2014, pp. 1-12.

12. Abramov S.A., Khmelnov D.E. Regular solutions of linear differential systems with power series coefficients. Programming and Computer Software, vol. 40, issue 2, 2014, pp. 98–10.

13. Abramov S.A., Ryabenko A.A., Khmelnov D.E. Procedures for searching local solutions of linear differential systems with infinite power series in the role of coefficients. Programming and Computer Software, vol. 42, issue 2, 2016, pp. 55–64.