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

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Numerical and Symbolic Scientific Computing

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Abstract

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 integro-differential 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.

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Acknowledgements

We acknowledge gratefully the support received from the SFB F013 in Subproject F1322 (principal investigators Bruno Buchberger and Heinz W. Engl), in earlier stages also Subproject F1302 (Buchberger) and Subproject F1308 (Engl). This support from the Austrian Science Fund (FWF) was not only effective in its financial dimension (clearly a necessary but not a sufficient condition for success), but also in a “moral” dimension: The stimulating atmosphere created by the unique blend of symbolic and numerical communities in this SFB – in particular the Hilbert Seminar mentioned in Sect. 1 – has been a key factor in building up the raw material for our studies.

Over and above his general role in the genesis and evolution of the SFB F1322, we would like to thank Heinz W. Engl for encouragement, critical comments and helpful suggestions, not only but especially in the early stages of this project.

Loredana Tec is a recipient of a DOC-fFORTE-fellowship of the Austrian Academy of Sciences at the Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz. Georg Regensburger was partially supported by the Austrian Science Fund (FWF): J 3030-N18.

We would also like to thank an anonymous referee for giving us plenty of helpful suggestions and references that certainly increased the value of this article.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Actuarial Science (SMSAS), University of Kent, Canterbury, CT2 7NF, UK

    Markus Rosenkranz

  2. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, 4040, Linz, Austria

    Georg Regensburger

  3. INRIA Saclay – Île de France, Project DISCO, L2S, Supélec, 91192, Gif-sur-Yvette Cedex, France

    Georg Regensburger

  4. Research Institute for Symbolic Computation (RISC), Johannes Kepler University, 4032, Hagenberg, Austria

    Loredana Tec & Bruno Buchberger

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  1. Markus Rosenkranz
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  4. Bruno Buchberger
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Correspondence to Markus Rosenkranz .

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  1. Inst. Numerische Mathematik, Universität Linz, Altenbergerstr. 69, Linz, 4040, Austria

    Ulrich Langer

  2. , Research Institute for Symbolic Computat, Johannes Kepler University, Altenbergerstr. 69, Linz, 4040, Austria

    Peter Paule

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Rosenkranz, M., Regensburger, G., Tec, L., Buchberger, B. (2012). Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases. In: Langer, U., Paule, P. (eds) Numerical and Symbolic Scientific Computing. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0794-2_13

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