TruncatedSeries Package

The TruncatedSeries package is designed to search for Laurent solutions, regular solutions and formal exponetial-logarithmic solutions of linear ordinary differential equations and systems with truncated series coefficients:

TruncatedSeries2023.zip – the archive with two files: maple.ind and maple.lib are a Maple library. Put these files to some directory, for example to ”/usr/userlib”. Assign

libname := ”/usr/userlib”, libname

in the Maple session.

The procedures LaurentSolution and RegularSolution have been implemented in Maple 2020 by D.E.Khmelnov. See for detail in

* Abramov S.A., Khmelnov D.E., Ryabenko A.A. Laurent solutions of linear ordinary differential equations with coefficients in the form of truncated power series. Computer Algebra: International Conference Proceedings, Moscow, June 17–21, 2019, pp.75–82.

• CA.Moscow.2019.Slides.pdf of the talk in Computer Algebra: Moscow, June 17–21, 2019

CA.Moscow.2019.Equation.mw – the Maple session with examples of the talk in Computer Algebra: Moscow, June 17–21, 2019.

CA.Moscow.2019.Equation.pdf – PDF version of the Maple session.

• Abramov S.A., Ryabenko A.A., Khmelnov D.E. Linear Ordinary Differential Equations and Truncated Series. Computational Mathematics and Mathematical Physics, 2019, Vol. 59, No. 10, pp. 1649–1659. (translation from С.А.Абрамов, А.А.Рябенко, Д.Е.Хмельнов. Линейные обыкновенные дифференциальные уравнения и усеченные ряды. Журнал вычислительной математики и математической физики, 2019, том 59, No 10, с. 1706–1717.)

• Abramov S.A., Ryabenko A.A., Khmelnov D.E. Regular Solutions of Linear Ordinary Differential Equations and Truncated Series. Computational Mathematics and Mathematical Physics, 2020, Vol. 60, No. 1, pp. 2–15. (translation from С.А.Абрамов, А.А.Рябенко, Д.Е.Хмельнов. Регулярные решения линейных обыкновенных дифференциальных уравнений и усеченные ряды. Журнал вычислительной математики и математической физики, 2020, том 60, No 1, с. 4–17.)

• 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. Programming and Computer Software, 2020, Vol. 46, No. 2, pp. 67–75. (translation from С.А.Абрамов, А.А.Рябенко, Д.Е.Хмельнов. Процедуры поиска лорановых и регулярных решений линейных дифференциальных уравнений с усеченными степенными рядами в роли коэффициентов. Труды ИСП РАН, 2019, том 31, вып. 5, с. 233–248)

TruncatedSeriesExamples.mw – the Maple session with examples of using the package for Laurent solutions and regular solutions.

TruncatedSeriesExamples.pdf – PDF version of the Maple session.

The procedure LaurentSolution for linear ordinary differential equations, each of the coefficients of which is either an algorithmically represented power series, or a truncated power series. It has been implemented in Maple 2020 by D.E.Khmelnov. See for detail in

• Abramov S.A., Khmelnov D.E., Ryabenko A.A. Truncated and Infinite Power Series in the Role of Coefficients of Linear Ordinary Differential Equations. CASC 2020. Lecture Notes in Computer Science, vol 12291, pp 63–76.

• The slides of the talk in CASC 2020

MixedSeriesExamples.mw – the Maple session with examples of using the package for Laurent solutions; samples from the paper and slides for CASC 2020.

MixedSeriesExamples.pdf – PDF version of the Maple session with examples of using the package for Laurent solutions.

The procedure FormalSolution has been implemented in Maple 2020 by A.A.Ryabenko. See for detail in

• S. A. Abramov, A. A. Ryabenko, and D. E. Khmelnov, Truncated Series and Formal Exponential-Logarithmic Solutions of Linear Ordinary Differential Equations. Computational Mathematics and Mathematical Physics, 2020, 60, p. 1609–1620 (translation from С.А.Абрамов, А.А.Рябенко, Д.Е.Хмельнов. Усеченные Ряды и Формальные Экспоненциально-Логарифмические Решения Линейных Обыкновенных Дифференциальных Уравнений. Журнал вычислительной математики и математической физики, 2020, том 60, No 10, с. 1664–1675.)

TruncatedSeriesFormalExamples2020.mw – the Maple session with examples of using the package for formal solutions; samples from the paper.

TruncatedSeriesFormalExamples2020.pdf – PDF version of the Maple session with examples of using the package for formal solutions.

• С.А.Абрамов, А.А.Рябенко, Д.Е.Хмельнов. Усеченные ряды. Труды XII Приокской научной конференции "Дифференциальные уравнения и смежные вопросы математики", 2020, Коломна, с. 8–19.

Kolomna2020.mw – the Maple session with examples from the paper.

Kolomna2020.pdf – PDF version of the Maple session.

The details will be in

• S. A. Abramov, A. A. Ryabenko, and D. E. Khmelnov, The TruncatedSeries Package for Solving Linear Ordinary Differential Equations Having Truncated Series Coefficients. In: Corless, R.M., Gerhard, J., Kotsireas, I.S. (eds) Maple in Mathematics Education and Research. MC 2020. Communications in Computer and Information Science, vol 1414, pp. 19–33 the talk in The 2020 Maple Conference.

TruncatedSeries_presentation.mw – the Maple session which is our presentation in the talk in the 2020 Maple Conference.

TruncatedSeries_presentation.pdf – PDF version of the Maple session.

• S. A. Abramov, A. A. Ryabenko, and D. E. Khmelnov, Procedures for Constructing Truncated Solutions of Linear Differential Equations with Infinite and Truncated Power Series in the Role of Coefficients. Programming and Computer Software, 2021, Vol. 47, No. 2, pp. 144–152 (translation from С.А.Абрамов, А.А.Рябенко, Д.Е.Хмельнов. Процедуры поиска усеченных решений линейных дифференциальных уравнений с бесконечными и усеченными степенными рялами в роли коэффициентов. Программирование, 2021, No 2, с. 56–65).

LaurentRegularExpLog.mw – the Maple session with examples from the paper.

LaurentRegularExpLog.pdf – PDF version of the Maple session.

Algorithms were previously proposed that allow one to find truncated solutions to linear differential equations with coefficients in the form of truncated formal power series. Below are suggested some automatic means of confirming the impossibility of obtaining a larger number of terms of such solutions without some additional information on a given equation. The confirmation has the form of a counterexample to the assumption about the possibility of obtaining some additional terms of the solution.

More details are to be found in:

• Khmelnov D.E., Ryabenko A.A., Abramov S.A. Automatic confirmation of exhaustive use of information on a given equation // Proceedings of 4th international conference "Computer Algebra", Moscow, 2021, pp.69–72

• The slides of the talk in Computer Algebra: Moscow, June 28–29, 2021

• Abramov S.A., Ryabenko A.A., Khmelnov D.E. Exhaustive use of information on an equation with truncated coefficients. Programming and Computer Software, 2022, Vol. 48, No. 2, pp. 116–124 (translation from С.А.Абрамов, А.А.Рябенко, Д.Е.Хмельнов. Исчерпывающее использование информации о дифференциальном уравнении с усеченными коэффициентами. Программирование, 2022, No 2, с. 63–72.).

ExhaustiveUseExamples.mw – the Maple session with examples for LaurentSolution and RegularSolution.

ExhaustiveUseExamples.pdf – PDF version of the Maple session.

• Abramov S.A., Khmelnov D.E., Ryabenko A.A., On truncated series involved in exponential-logarithmic solutions of truncated LODEs. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2022. Lecture Notes in Computer Science, vol 13366, pp. 18–28

• The slides of the talk in CASC 2022

ExhaustiveUse_FormalSolution.mw – the Maple session with examples for FormalSolution of the talk in CASC 2022.

ExhaustiveUse_FormalSolution.pdf – PDF version of the Maple session.

The procedure LaurentSolution has been extended for the case of systems with all coefficients given as truncated series.

We consider systems of linear ordinary differential equations with infinite formal power series as coefficients. The series are represented in a truncated form, while the degree of truncation may differ for different coefficients. As a tool of studying such systems we use induced recurrent systems and literal designations for unspecified coefficients of the series. For the case when the determinant of the leading matrix of the induced system is not zero and does not contain literals, an algorithm for constructing Laurent solutions of the system is proposed. The series included in the solutions are still truncated. The algorithm finds for them the maximum possible number of terms that are invariant with respect to any prolongations of truncated coefficients of the original system. The implementation of the algorithm as a Maple procedure and examples of its usage are presented.

See examples of the procedure's use for the systems in

LaurentForSystem2022.mw – the Maple session.

LaurentForSystem2022.pdf – PDF version of the Maple session.

In the second version of LaurentSolution for truncated system, we advance in extending our algorithm to the case when the leading matrix of the induced system is singular using algorithm EG as an auxiliary tool. More details are to be found in:

• Khmelnov D.E., Ryabenko A.A. Algorithm EG as a Tool for Finding Laurent Solutions of Linear Differential Systems with Truncated Series Coefficients // Computer algebra: 5th International Conference Materials. Moscow, 26–28 June, 2023/ ed. S.A. Abramov, A.B. Batkhin, L.A. Sevastyanov. Moscow: KIAM, 2023, pp.92–96

• The slides of the talk in Computer Algebra: Moscow, June 26-28, 2023

See examples in

LaurentForSystem2023.mw – the Maple session.

LaurentForSystem2023.pdf – PDF version of the Maple session.