## Section: New Results

### Boundary value problems for linear ordinary integro-differential equations

Participants : Alban Quadrat, Georg Regensburger.

In [35] , we study algorithmic aspects of linear ordinary
integro-differential operators with polynomial coefficients. Even
though this algebra is not noetherian and has zero divisors, Bavula
recently proved in [85] that it is coherent, which
allows one to develop an algebraic systems theory. For an algorithmic
approach to linear systems theory of integro-differential equations
with boundary conditions, computing the kernel of matrices is a
fundamental task. As a first step, we have to find annihilators, which
is, in turn, related to polynomial solutions. We present an
algorithmic approach for computing polynomial solutions and the index
for a class of linear operators including integro-differential
operators. A generating set for right annihilators can be constructed
in terms of such polynomial solutions. For initial value problems, an
involution of the algebra of integro-differential operators also
allows us to compute left annihilators, which can be interpreted as
compatibility conditions of integro-differential equations with
boundary conditions. These results are implemented in Maple
based on the `IntDiffOp` and `IntDiffOperations`
packages.