The algebra of integro-differential operators on a polynomial algebra

Bavula, V. V.

doi:10.1112/jlms/jdq081

Cited by 29 publications

(106 citation statements)

References 52 publications

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“…Finally, in [1,2,3], various algebraic properties of I and important results are proven amongst them a classification of simple modules, an analogue of Stafford's theorem, and of the first conjecture of Dixmier.…”

Section: Let Us Consider the Following Inhomogeneous Id Equationmentioning

confidence: 99%

“…The algebra I(k), simply be denoted by I in what follows, was studied in [1,3] as a generalized Weyl algebra. See [20] for the construction of I as a factor algebra of a skew polynomial ring.…”

Section: Ordinary Integro-differential Operators With Polynomial Coefmentioning

confidence: 99%

“…In contrast to [1,3], this last approach allows one to have more than one point evaluation, which is crucial for the study of boundary value problems.…”

Section: Let Us Consider the Following Inhomogeneous Id Equationmentioning

confidence: 99%

“…Based on the normal forms for generalized Weyl algebras, it is shown in [3] that I admits the involution θ…”

Section: Let Us Consider the Following Inhomogeneous Id Equationmentioning

confidence: 99%

“…Algebras of ordinary ID operators have recently been studied within an algebraic approach in [1,2,3,4] and within an algorithmic approach in [20,21,22]. The goal of the latter works is to provide an algebraic and algorithmic framework for studying boundary value problems and Green's operators.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial Solutions and Annihilators of Ordinary Integro-Differential Operators

Quadrat

Regensburger

2013

IFAC Proceedings Volumes

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In this paper, we study algorithmic aspects of linear ordinary integro-differential operators with polynomial coefficients. Even though this algebra is not noetherian and has zero divisors, Bavula recently proved that it is coherent, which allows one to develop an algebraic systems theory. For an algorithmic approach to linear systems theory of integro-differential equations with boundary conditions, computing the kernel of matrices is a fundamental task. As a first step, we have to find annihilators, which is, in turn, related to polynomial solutions. We present an algorithmic approach for computing polynomial solutions and the index for a class of linear operators including integro-differential operators. A generating set for right annihilators can be constructed in terms of such polynomial solutions. For initial value problems, an involution of the algebra of integro-differential operators also allows us to compute left annihilators, which can be interpreted as compatibility conditions of integro-differential equations with boundary conditions. We illustrate our approach using an implementation in the computer algebra system Maple. Finally, system-theoretic interpretations of these results are given and illustrated on integro-differential equations.

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Section: Let Us Consider the Following Inhomogeneous Id Equationmentioning

confidence: 99%

Section: Ordinary Integro-differential Operators With Polynomial Coefmentioning

confidence: 99%

“…In contrast to [1,3], this last approach allows one to have more than one point evaluation, which is crucial for the study of boundary value problems.…”

Section: Let Us Consider the Following Inhomogeneous Id Equationmentioning

confidence: 99%

“…Based on the normal forms for generalized Weyl algebras, it is shown in [3] that I admits the involution θ…”

Section: Let Us Consider the Following Inhomogeneous Id Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Polynomial Solutions and Annihilators of Ordinary Integro-Differential Operators

Quadrat

Regensburger

2013

IFAC Proceedings Volumes

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Computing Polynomial Solutions and Annihilators of Integro-Differential Operators with Polynomial Coefficients

Quadrat

Regensburger

2020

Advances in Delays and Dynamics

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In this paper, we study algorithmic aspects of the algebra of linear ordinary integro-differential operators with polynomial coefficients. Even though this algebra is not Noetherian and has zero divisors, Bavula recently proved that it is coherent, which allows one to develop an algebraic systems theory over this algebra. For an algorithmic approach to linear systems of integro-differential equations with boundary conditions, computing the kernel of matrices with entries in this algebra is a fundamental task. As a first step, we have to find annihilators of integro-differential operators, which, in turn, is related to the computation of polynomial solutions of such operators. For a class of linear operators including integro-differential operators, we present an algorithmic approach for computing polynomial solutions and the index. A generating set for right annihilators can be constructed in terms of such polynomial solutions. For initial value problems, an involution of the algebra of integro-differential operators then allows us to compute left annihilators, which can be interpreted as compatibility conditions of integro-differential equations with boundary conditions. We illustrate our approach using an implementation in the computer algebra system Maple.

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On Symbolic Approaches to Integro-Differential Equations

Boulier

Lemaire

Rosenkranz

et al. 2020

Advances in Delays and Dynamics

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Recent progress in computer algebra has opened new opportunities for the parameter estimation problem in nonlinear control theory, by means of integrodifferential input-output equations. This paper recalls the origin of integro-differential equations. It presents new opportunities in nonlinear control theory. Finally, it reviews related recent theoretical approaches on integro-differential algebras, illustrating what an integro-differential elimination method might be and what benefits the parameter estimation problem would gain from it.

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The algebra of integro-differential operators on a polynomial algebra

Abstract: We prove that the algebra In := K x1, .

Cited by 29 publications

References 52 publications

Polynomial Solutions and Annihilators of Ordinary Integro-Differential Operators

Polynomial Solutions and Annihilators of Ordinary Integro-Differential Operators

Computing Polynomial Solutions and Annihilators of Integro-Differential Operators with Polynomial Coefficients

On Symbolic Approaches to Integro-Differential Equations

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