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.

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. The main goals of this project were to investigate algorithmic and algebraic methods for integro-differential algebras and for linear systems of integro-differential equations. Implementing the algorithms developed in computer algebra systems, was also an important aspect of the project. We applied our new methods and software to problems in control theory, matrix theory, quantum physics, and chemical reaction networks. Many processes in science and engineering can be modeled by linear systems of differential, delay, and integral equations. For analyzing such systems, one usually computes with the corresponding matrices and linear operators. Instead of working with actual matrices and operators, symbolic computation works with symbols representing mathematical objects. To implement symbolic computations with operators on a computer, one needs a unique way of representing them. In the project, we developed a new method to find and to prove normal forms of differential, delay, and integral operators with matrix coefficients, for instance. We used such normal forms to automatize and generalize computations for linear differential time-delay systems in control theory. If input and output of operators or matrices have different dimensions, they cannot be added and composed arbitrarily. This restricts valid computations with operators and matrices. In the project, we developed a new algebraic framework for this situation. The idea is to first compute symbolically without restrictions and then justify the result independent of how it was obtained. With our collaborators, we applied this approach and our software to obtain computer-assisted proofs for new results in the theory of generalized inverses. The long-term goal is to provide a comprehensive framework and software support for automated proofs of properties of linear operators in various research areas. In a collaboration with theoretical physicists, we used normal forms of nested integrals in the analysis of certain processes in quantum electrodynamics and quantum chromodynamics. These computations are crucial for precise evaluation of measurement data collected at particle colliders. The software for performing the computation is based on theoretical results on the algebraic structure of certain integro-differential rings. In the framework of the project, we also continued our research on dynamical systems arising from chemical reaction networks and positive solutions of the corresponding polynomial equations. The new methods based on sign vectors and results on positive steady states opened promising research agendas for fundamental questions in real algebraic geometry as well as for bioinformatics and bioengineering.