From Ordinary to Partial Integral Operators

From Ordinary to Partial Integral Operators

The solution of a boundary problem is typically described by an integral operator known as its Green's operator, both for ordinary and for partial differential equations. Using a symbolic description of integral operators, some Green's operators should be representable in closed form, others decomposed into simpler ones. We have worked a suitable approach along these lines for the case of ordinary differential equations. In this talk, we shall first review the key results for the ordinary case and then present the first steps for tackling the partial case. Modelling the operator algebras by noncommutative polynomials (with infinitely many indeterminates), the crucial tool for obtaining a constructive setup is noncommutative Groebner bases (for an infinitely generated ideal of relations).