Introduction

My research has involved the development of algorithms to determine properties of systems of nonlinear partial differential equations. Its current emphasis is on the development of symbolic-numeric algorithms. My research area lies at the intersection of algebraic, geometric and algorithmic methods for differential equations; with computer experimentation spurring theoretical developments, and theory prompting experiment.

Applications include algorithmic determination of canonical forms for systems of differential equations which are suitable for their analytical or numerical solution (for example, their solution using directed graphs and the determination of all solutions lying in some finite extension of the rational function field). Another application of my work has been the design of algorithms which can calculate the dimension and structure of Lie symmetry algebras of differential equations. Previous methods for calculating structure depended on integration, a process that is not always guaranteed to succeed. Our Maple implemented programs and extensive documentation are publically available.

My most significant recent contributions have been the development of a effective canonical form algorithm for systems of polynomially nonlinear systems of partial differential equations and an effective algorithm for determining the structure of infinite Lie pseudogroups.