Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gr\"obner Bases

TL;DR

This paper develops an algebraic framework for solving linear boundary problems in differential equations using integro-differential algebras, canonical forms, and symbolic computation, with implementation in the Theorema system.

Contribution

It introduces integro-differential algebras and operators, providing a new algebraic approach and automated tools for boundary problem analysis and solution.

Findings

Algebraic structures enable symbolic boundary problem solving.

Canonical forms facilitate automated proofs and simplifications.

Implementation demonstrates practical applicability in Theorema.

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.