Symbolic Computation and Differential and Difference Equations

North Carolina State University, Raleigh NC

Investigators

Abstract

This research addresses foundational and computational issues concerning the algebraic behavior of systems of linear difference and differential equations and allows researchers to understand aspects of the qualitative behavior of solutions of these equations. This research develops algorithms that will be of use not only to mathematicians but engineers and scientists in many fields as well. These algorithms will be the basis of code appearing in symbolic computation computer packages used in the education as well as the day-to-day work of engineers and other scientists. In particular, this research investigates new and more efficient algorithms for the classical Picard-Vessiot theories of over determined systems of differential and difference equations, and develops new algorithms to determine elementary algebraic properties of under determined systems of linear partial differential and difference equations. It also extends the classical Picard-Vessiot theory and its algorithms to Galois theories with infinite dimensional groups, allowing applications to parameterized linear equations as well as some classes of nonlinear equations.