A skew polynomial approach to integro-differential operators

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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“…See [21,20] for details, in particular, for a Gröbner basis of the defining relations. For α = 0, a boundary operator (8) is of the form…”

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%

Polynomial Solutions and Annihilators of Ordinary Integro-Differential Operators

Quadrat

Regensburger

2013

IFAC Proceedings Volumes

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“…In any integro-differential algebra, one can define an evaluation E = 1 − ∂ , which corresponds to f (a) in the above example. See [21] for further examples and [19] for a short summary.…”

Section: Integro-differential Operators and Polynomialsmentioning

confidence: 99%

An Automated Confluence Proof for an Infinite Rewrite System Parametrized over an Integro-Differential Algebra

Tec

Regensburger

Rosenkranz

et al. 2010

Mathematical Software – ICMS 2010

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“…See Section 5 for more details and an implementation. Alternatively, integro-differential operators can also be defined directly in terms of normal forms [20].…”

Section: Representation Of Integro-differential Operatorsmentioning

confidence: 99%

A Symbolic Framework for Operations on Linear Boundary Problems

Rosenkranz

Regensburger

Tec

et al. 2009

Computer Algebra in Scientific Computing

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Abstract. We describe a symbolic framework for treating linear boundary problems with a generic implementation in the Theorema system. For ordinary differential equations, the operations implemented include computing Green's operators, composing boundary problems and integrodifferential operators, and factoring boundary problems. Based on our factorization approach, we also present some first steps for symbolically computing Green's operators of simple boundary problems for partial differential equations with constant coefficients. After summarizing the theoretical background on abstract boundary problems, we outline an algebraic structure for partial integro-differential operators. Finally, we describe the implementation in Theorema, which relies on functors for building up the computational domains, and we illustrate it with some sample computations including the unbounded wave equation.

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A skew polynomial approach to integro-differential operators

Cited by 31 publications

References 28 publications

Polynomial Solutions and Annihilators of Ordinary Integro-Differential Operators

Polynomial Solutions and Annihilators of Ordinary Integro-Differential Operators

An Automated Confluence Proof for an Infinite Rewrite System Parametrized over an Integro-Differential Algebra

A Symbolic Framework for Operations on Linear Boundary Problems

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