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. For an algorithmic approach to linear systems, computing linear relations (syzygies) of integro-differential operators is a crucial task. Bavula proved recently that if one starts with finitely many ordinary integro- differential equations with polynomial coefficients and initial conditions, the related compatibility conditions and relations can in principle be described in finite terms. Based on our approach for computing annihilators and polynomial solutions, we want to develop constructive methods for syzygies and systems of ordinary integro-differential equations with boundary conditions. More generally, we will study coherent operator algebras on univariate polynomials for which syzygies can effectively be computed. A central idea of algebraic systems theory is to associate to a linear system defined by a matrix of operators a module that captures the main properties of the solution space. In the proposed project, we want to investigate (constructive) methods from algebraic systems theory for systems of ordinary integro-differential equations. Boundary value linear systems in mathematical systems and control theory are an important class of such systems, which we will study. We will also develop further our algebraic and algorithmic methods for linear ordinary and partial boundary problems and (generalized) Green`s operators and the corresponding software. Implementing the constructive methods to be developed in computer algebra systems, is also an important aspect of the project.