  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.
