Algebra and algorithms for integro-differential equations

Integro-differential equations and boundary (value) problems are ubiquitous in science, engineering, and applied mathematics. While algebraic structures and computer algebra for differential equations per se are very well developed, the investigation of their integro-differential counterparts has started only recently. We developed with our co-authors a symbolic computation approach for linear ordinary boundary problems and their Green's (solution) operators. It is based on integro-differential operators over integro-differential algebras, allowing to compute with boundary problems (differential operator plus boundary conditions) as well as Green's operators (integral operators) in a single algebraic structure. The goal of the proposed project is to investigate algorithmic and algebraic methods for linear systems of integro-differential equations with boundary conditions, complementing numerical methods. We will study computable integro-differential algebras whose elements can be represented in a computer and algebraic properties of the associated integro-differential operators. In particular, we want to develop symbolic methods for computing rational and computable classes of solutions and the corresponding compatibility conditions for inhomogeneous equations.