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Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Groebner Bases

Rosenkranz, Markus and Regensburger, Georg and Tec, Loredana and Buchberger, Bruno (2012) Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Groebner Bases. In: Langer, Ulrich and Paule, Peter, eds. Numerical and Symbolic Scientific Computing: Progress and Prospects. Texts and Monographs in Symbolic Computation . Springer, Wien, pp. 273-331. ISBN 0943853X. (doi:10.1007/978-3-7091-0794-2_5) (KAR id:26236)

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We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integrodifferential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the THEOREMA system; some code fragments and sample computations are included.

Item Type: Book section
DOI/Identification number: 10.1007/978-3-7091-0794-2_5
Projects: [152] F1322
Uncontrolled keywords: Symbolic computation; computer algebra; symbolic analysis; differential equations; boundary problems
Subjects: Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Q Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations
Q Science > QA Mathematics (inc Computing science) > QA 76 Software, computer programming,
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Markus Rosenkranz
Date Deposited: 17 Feb 2011 15:01 UTC
Last Modified: 16 Nov 2021 10:04 UTC
Resource URI: (The current URI for this page, for reference purposes)
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