A Symbolic Framework for Operations on Linear Boundary Problems

Rosenkranz, Markus; Regensburger, Georg; Tec, Loredana; Buchberger, Bruno

doi:10.1007/978-3-642-04103-7_24

Cited by 19 publications

(35 citation statements)

References 22 publications

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“…Interestingly, their systematic treatment in Symbolic Computation started rather recently [17]. For handling the central problems of solving and factoring boundary problems, a differential algebra setting for LODEs is employed in [20,19] and for LPDEs in [21,18]. An overarching abstract framework based only on Linear Algebra is developed in [15].…”

Section: Introductionmentioning

confidence: 99%

Green's Functions for Stieltjes Boundary Problems

Rosenkranz

Serwa

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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Abstract. Stieltjes boundary problems generalize the customary class of well-posed two-point boundary value problems in three independent directions, regarding the specification of the boundary conditions: (1) They allow more than two evaluation points. (2) They allow derivatives of arbitrary order. (3) Global terms in the form of definite integrals are allowed. Assuming the Stieltjes boundary problem is regular (a unique solution exists for every forcing function), there are symbolic methods for computing the associated Green's operator.In the classical case of well-posed two-point boundary value problems, it is known how to transform the Green's operator into the so-called Green's function, the representation usually preferred by physicists and engineers. In this paper we extend this transformation to the whole class of Stieltjes boundary problems. It turns out that the extension (1) leads to more case distinction, (2) implies ill-posed problems and hence distributional terms, (3) has apparently no effect on the structure of the Green's function.

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Section: Introductionmentioning

confidence: 99%

Green's Functions for Stieltjes Boundary Problems

Rosenkranz

Serwa

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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“…Methods for solving and factoring boundary problems are described in [18,22], both in a differential algebra and in an abstract setting. See also [23] for an overview of operations on linear boundary problems in a symbolic framework. The integro-differential operators are realized by a suitable quotient of noncommutative polynomials over a given integro-differential algebra.…”

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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“…The other one is a more faithful abstraction of the integration-by-parts formula (see Eq. (7)), giving rise to the concept of an integro-differential algebra [37] which has generated much interest [1,2,38,39]. As suggested in previous cases, free objects for these algebraic structures are important in their studies.…”

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confidence: 99%

Constructions of Free Commutative Integro-Differential Algebras

Gao

Guo

2014

Lecture Notes in Computer Science

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In this survey, we outline two recent constructions of free commutative integro-differential algebras. They are based on the construction of free commutative Rota-Baxter algebras by mixable shuffles. The first is by evaluations. The second is by the method of Gröbner-Shirshov bases. 1 2 XING GAO AND LI GUO The study of differential algebra began with Ritt's classic work [35,36]. After the fundamental work of Kolchin [33], differential algebra has evolved into a vast area of mathematics that is important in both theory [15,44] and applications: for instance, in mechanic theorem proving by W.-T. Wu [45,46]. Free (commutative) differential algebras, in the form of differential polynomial algebras (Theorem 2.3), are essential for studying differential equations, as polynomial algebras are for commutative algebras.The algebraic study of integrals came much later. In fact the development did not start from an algebraic abstraction of integrals, but from the effort of G. Baxter [5] in 1960 to understand a formula in probability theory. As a result, the concept is not called an integral algebra, but called a (Rota-)Baxter algebra (Eq. (3)) which is the integral counterpart of the derivation, the difference operator, and divided differences (see Eq. (1)). Soon afterwards Rota noticed its importance in combinatorics and promoted its study through research and survey articles (see e.g. [40,41]). Independently, Rota-Baxter operators on Lie algebras were found to be closely related to the classical Yang-Baxter equation [42]. Since the turn of this century, the theory of Rota-Baxter algebra has experienced rapid development with broad applications in mathematics and physics [4,21,29,40,41,42], especially noteworthy in the Hopf algebra approach of Connes-Kreimer to renormalization of quantum field theory [16,19,29]. Here again a fundamental role is played by free (commutative) Rota-Baxter algebras that were first constructed by Rota [40] and Cartier [14], and then by in terms of mixable shuffles (Theorem 2.4).The fusion of differential and Rota-Baxter algebras, motivated by algebraic study of calculus as a whole, appeared about five years ago. It is amazing that two structures for this purpose were introduced at about the same time. One is a relatively simple coupling of differential algebra and Rota-Baxter algebra through section axiom (Eq. (4)) that reflects the First Fundamental Theorem of Calculus. It is called differential Rota-Baxter algebra [26]. The other one is a more faithful abstraction of the integration-by-parts formula (see Eq. (7)), giving rise to the concept of an integro-differential algebra [37] which has generated much interest [1,2,38,39]. As suggested in previous cases, free objects for these algebraic structures are important in their studies. Because of the relative independence of the differential and integral (Rota-Baxter) structures in a differential Rota-Baxter algebra, the free object was constructed by a clear combination of the free objects on the differential and Rota-Baxter sides and were obtai...

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A Symbolic Framework for Operations on Linear Boundary Problems

Cited by 19 publications

References 22 publications

Green's Functions for Stieltjes Boundary Problems

Green's Functions for Stieltjes Boundary Problems

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

Constructions of Free Commutative Integro-Differential Algebras

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