The LFS package is for solving Linear Functional (differential, difference and q-difference) Systems of equations.
The procedure RationalSolution is an implementation in Maple 2020 of two algorithms to construct a basis of rational-function solutions for homogeneous systems with rational-function coefficients and a particular solution for inhomogeneous systems with rational-function right-hand sides. See for detail in the slides of the talk S. Abramov, D.Khmelnov, A.Ryabenko «Searching for rational solutions of linear ordinary difference and differential systems: schemes equipped with checkpoints» in The conference "Программирование и вычислительная математика" памяти Н. П. Трифонова (1925–2020). See also the abstract of this talk in pages 12-13 of Программирование и вычислительная математика. тезисы докладов конференции памяти Н. П. Трифонова / под ред. С. А. Абрамова, А. В. Столярова. – Москва : МАКС Пресс, 2020.
The procedure HypergeometricSolution is an implementation in Maple 2020 of an algorithm to construct a particular hypergeometric solutions for inhomogeneous systems with rational-function coefficients and hypergeometric right-hand sides. See for detail in Ryabenko A.A. Particular Solutions of Linear Differential and (q-)Difference Systems with Hypergeometric Right-hand Sides // Programming and Computer Software. 2019. Vol. 45. No 5. P. 298-302
The procedure LaurentSolution is a draft implementation in Maple2023 of an algorithm to construct Laurent series solutions for homogeneous differential systems with polynomial or power series coefficients (implemented by A.A.Ryabenko, D.E.Khmelnov).
lfs.zip – the archive with two files: lfs.ind and lfs.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 matrix form of a full-rank system:
An(x) ξ^n(y(x)) + … + A1(x) ξ(y(x)) + A0(x) y(x) = b(x),
where
The normal form of a first order system:
ξ(y(x)) = A(x) y(x) + b(x),
where
For the differential case: ξ(y(x)) = y'(x).
For the difference case: ξ(y(x)) = y(x+1).
For the q-difference case: ξ(y(x)) = y(q x), where q is a name or an integer large then 1.
essay_on_laurent_solution.mw - сессия Maple 2023 с примерами из Эссе о лорановых решениях линейных дифференциальных систем
computer_algebra_for_operator_matrices.mw - сессия Maple 2024 с примерами из Абрамов С.А., Рябенко А.А., Хмельнов Д.Е., Компьютерная алгебра операторных матриц, Москва : МАКС Пресс, 2024, https://doi.org/10.29003/m3796.978-5-317-07135-6.
lfs_laurentsolution_examples.mw - the Maple 2023 session file with Laurent solution examples.
lfs_laurentsolution_examples.pdf - the pdf copy of that Maple session.
The optional argument of RationalSolution and HypergeometricSolution may be
The default is 'earlyterminate' = true.
lfs_rational_examples.mw - the Maple session file with rational solutions examples.
lfs_rational_examples.pdf - the pdf copy of that Maple session.
lfs_examples.mw - the Maple session file with hypergeometric examples.
lfs_examples.pdf - the pdf copy of that Maple session.