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

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.

54 pages

References in corpus (3)

• The algebra of integro-differential operators on a polynomial algebra
• Finite Groebner bases in infinite dimensional polynomial rings and applications
• Equivariant Groebner bases and the Gaussian two-factor model

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