Integro-Differential Operators and Algebraic Systems Theory

Boundary (value) problems play a dominant role in applied mathematics since in practical problems differential equations usually come along with boundary conditions. Despite this fact, they have not been considered much from a symbolic computation perspective, unlike differential equations per se. In algebraic systems theory the situation is in some respect similar. Many questions on linear systems for ordinary/partial differential, shift, and time-delay operators can now be treated in a uniform way and answered constructively. This was made possible by combining symbolic methods for certain operator algebras (namely Ore algebras) with techniques from homological algebra. But this approach cannot deal with boundary conditions since they are not expressible within this algebraic setting. On the other hand, systems with boundary conditions of course play an important role in mathematical systems theory in general and in applications. Our research goal is to provide an algebraic foundation and algorithmic framework for solving, transforming and simplifying boundary problems, complementing numerical methods. For boundary problems with linear ordinary differential equations we have introduced the structure of integro-differential algebra that combines a differential algebra with suitable notions of integration and evaluation. The ring of integro-differential operators associated with an ordinary integro-differential algebra allows us to express and compute with boundary problems (differential operator plus boundary conditions) as well as solution operators (Green`s operators) in one algebraic structure. In short, the goal of the proposed project is to combine both approaches and study algebraic and algorithmic aspects of integro-differential operators in the context and by methods of algebraic systems theory. For this we can in principle use the generic techniques from algebraic systems theory via homological algebra. But the existing constructive methods for Ore algebras cannot be applied directly since integro-differential operators do not fit in this class. So the main questions are: What are the relevant algebraic properties of integro-differential operators? Which constructions on matrices of integro-differential operators and more generally from homological algebra can be made algorithmic? How do properties of systems with boundary conditions translate to properties of the corresponding algebraic objects? Integro-differential operators will allow us to treat linear one-dimensional systems of (integro-)differential equations (time-invariant and time-varying) with boundary conditions by algebraic and symbolic methods. For the constructive methods to be developed in the project, we will work on prototype implementations in a computer algebra system, and we will also study our approach on problems coming from applications.