Solving and factoring boundary problems for linear ordinary differential equations in differential algebras
type: Item
instance of: scholarly article
title: Solving and factoring boundary problems for linear ordinary differential equations in differential algebras
zbMATH Open document ID: 1151.34008
author: Markus Rosenkranz
author: Georg Regensburger
published in: Journal of Symbolic Computation
publication date: 2008-06-11T00:00:00Z
review text: We present a new approach for expressing and solving boundary problems for linear ordinary differential equations in the language of differential algebras. Starting from an algebra with a derivation and integration operator, we construct an algebra of linear integro-differential operators that is expressive enough for specifying regular boundary problems with arbitrary Stieltjes boundary conditions as well as their solution operators. On the basis of these structures, we define a new multiplication on regular boundary problems in such a way that the resulting Green's operator is the reverse composition of the constituent Green's operators. We also provide a method for lifting any factorization of the underlying differential operator to the level of boundary problems. Since this method only needs the computation of initial value problems, it can be used as an effective alternative for computing Green's operators in the case where one knows how to factor the given differential operators.
reviewed by: Mykola Grygorenko
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 5287781
zbMATH Keywords: Linear boundary value problems
zbMATH Keywords: Ordinary differential equations
zbMATH Keywords: Green's operators
zbMATH Keywords: Factorization
zbMATH Keywords: Differential algebra
zbMATH Keywords: Noncommutative Gröbner bases
MaRDI profile type: Publication
full work available at URL: j.jsc.2007.11.007
OpenAlex ID: W1999837891
cites work: An analytic problem whose solution follows from a simple algebraic identity
cites work: The diamond lemma for ring theory
cites work: Duality Theory fornth Order Differential Operators under Stieltjes Boundary Conditions
cites work: Ordinary Differential Operators Under Stieltjes Boundary Conditions
cites work: Q4148232
cites work: Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. Translation from the German
cites work: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems
cites work: Q4394946
cites work: Q4377762
cites work: On the Nature of the Spectrum of Singular Second Order Linear Differential Equations
cites work: Q3937313
cites work: Q4782270
cites work: Q4132170
cites work: Q3496518
cites work: New extremal characterizations of generalized inverse of linear operators
cites work: Factoring and solving linear partial differential equations
cites work: Generalized Loewy-decomposition of d-modules
cites work: Loewy and primary decompositions of \(\mathcal D\)-modules
cites work: Complexity of factoring and calculating the GCD of linear ordinary differential operators
cites work: Q3153360
cites work: Q3255437
cites work: On the ring of hurwitz series
cites work: Hurwitz series as formal functions
cites work: Q5680222
cites work: Q5543730
cites work: Germs and their Operational Calculus
cites work: Q3808248
cites work: An introduction to commutative and noncommutative Gröbner bases
cites work: A unified approach to generalized inverses of linear operators: I. Algebraic, topological and projectional properties
cites work: Q5431990
cites work: An algebraic foundation for factoring linear boundary problems
cites work: Local Differential Algebra
cites work: A new symbolic method for solving linear two-point boundary value problems on the level of operators
cites work: Solving Linear Boundary Value Problems Via Non-commutative Gröbner Bases
cites work: Baxter algebras and combinatorial identities. I
cites work: Q3853967
cites work: Q4227347
cites work: Factorization of linear partial differential operators and Darboux integrability of nonlinear PDEs
cites work: Q4394959
cites work: Q4790565
DOI: 10.1016/J.JSC.2007.11.007
description: scientific article; zbMATH DE number 5287781