Solutions for linear functional systems

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

• Ai(x) are matrices whose entries are rational functions or power series of x;
• b(x) is a column vector of finite sums of hypergeometric terms of x;
• y(x) is a column vector of unknown functions;
• ξ is a pseudo-linear map.

The normal form of a first order system:

ξ(y(x)) = A(x) y(x) + b(x),

where

• A(x) is a matrix whose entries are rational functions in x.

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

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

• 'earlyterminate' = true, the rational solution's algorithm with checkpoints is used, the algorithms from Abramov S.A. When the Search for Solutions Can Be Terminated // In: Pillwein V., Schneider C. (eds) Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra. In Honour of Peter Paule on his 60th Birthday.Springer, Cham. 2020. P. 1–7. and from Abramov S.A., Khmelnov D.E., Ryabenko A.A. Checkpoints in searching for rational solutions of linear ordinary difference and differential systems // ACM Communications in Computer Algebra. 2020. Vol. 54. Issue 2. P. 18–29 are used.

• 'earlyterminate' = false, the algorithm from Abramov S.A., Khmelnov D.E. Denominators of rational solutions of linear difference systems of an arbitrary order. Progr. and Comp. Softwr, No 2, 2012, pp. 84-91 is used.

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.