AF: Small: Algorithmic Algebraic Methods for Systems of Difference-Differential Equations

Catholic University Of America, Washington DC

Investigators

Abstract

Differential, difference and difference-differential equations constitute main tools that scientists and engineers use to create mathematical models of real-life phenomena. Whereas continuous-time and discrete-time processes depending on several factors are described by systems of partial differential and difference equations, respectively, processes that include both continuous and discrete components (such as processes with time delay caused by the time required to transport mass, energy or information) are governed by systems of partial difference-differential equations (PDDEs). Furthermore, very often characteristics of physical, chemical or biological processes have certain symmetries, which can be captured mathematically as transformation group actions. Thus, the development of computational methods and algorithms for systems of PDDEs and such systems with group action is of primary importance in applications. Despite the over sixty-year history of constructive methods in differential and difference algebra, there are currently no efficient computational techniques for algebraic PDDEs. This project aims to develop the theory, methods and algorithms to determine the structure of solutions of systems of such equations including algebraic PDDEs with symmetry group actions. The research results will be applied to systems that describe mathematical models in physics, chemistry and biology. The educational goal of the project is to create an interdepartmental program on applications of symbolic computation that will involve undergraduate and graduate majors in computer science, mathematics, physics and biology at the Catholic University of America (CUA). The key research directions of this project are as follows. (1) Development of computational methods and algorithms for difference-differential elimination and for decomposition of solution sets of systems of algebraic PDDEs into unions of simple components. Extension of the obtained techniques to systems with group actions and/or weighted operators. (2) Development of algorithms for building Groebner-type bases in difference-differential modules and algebras. Applications of these algorithms to the computation of dimension functions of algebraic PDDEs that arise in applications. (3) Consistency analysis of finite difference approximations of algebraic differential equations via the techniques of generalized Groebner bases and difference-differential characteristic sets. (4) Application of the obtained methods and algorithms to systems of PDDEs that play fundamental roles in physics, engineering, chemical and biological modeling. The main methods and approaches of the project include the techniques of generalized difference-differential characteristic sets and relative Groebner bases, the use of dimension polynomials and quasi-polynomials, and decomposition methods for systems of algebraic PDDEs and such systems with group action and weighted basic operators. The results will be demonstrated in interdisciplinary research projects at CUA. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.