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
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.
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
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
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.
Kolomna2020.mw – the Maple session with examples from the paper.
Kolomna2020.pdf – PDF version of the Maple session.
The details will be in
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.
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:
ExhaustiveUseExamples.mw – the Maple session with examples for LaurentSolution and RegularSolution.
ExhaustiveUseExamples.pdf – PDF version of the Maple session.
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:
See examples in
LaurentForSystem2023.mw – the Maple session.
LaurentForSystem2023.pdf – PDF version of the Maple session.