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

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 TH∃OREM∀ system; some code fragments and sample computations are included. Markus Rosenkranz School of Mathematics, Statistics and Actuarial Science (SMSAS), University of Kent, Canterbury CT2 7NF, United Kingdom, e-mail: [email protected] Georg Regensburger Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, 4040 Linz, Austria, INRIA Saclay – Île de France, Project DISCO, L2S, Supélec, 91192 Gif-sur-Yvette Cedex, France, e-mail: [email protected] Loredana Tec Research Institute for Symbolic Computation (RISC), Johannes Kepler University, 4032 Castle of Hagenberg, Austria, e-mail: [email protected] Bruno Buchberger Research Institute for Symbolic Computation (RISC), Johannes Kepler University, 4032 Castle of Hagenberg, Austria, e-mail: [email protected]