Partial Differential Equations, Ordinary Differential Equations and Stochastic Partial Differential Equation models in science and engineering

The research covers:
Function spaces (distributions, Sobolev spaces, etc)
Existence, uniqueness, regularity and singular properties of solutions
Singularly perturbed problems
Numerical treatment by finite element, finite difference and boundary element methods
Dynamical systems
Interval methods
Modification of mathematical models
Nonlinear theories of generalized functions

Study of dense singularities of solutions of nonlinear PDEs

The emphasis is placed on:
Lie semigroups of noninvertible transformations of solutions
Abstract differential geometry of algebras of generalised functions and de Rham cohomology
Space-time foam structures with dense singularities

Homogenization of elliptic and evolution problems

Geometric partial differential equations

The focus is on:
Harmonic and wave maps on Riemann-Finsler manifolds
Ricci flow on Finsler manifolds

Mathematical biology with emphasis on epidemiology

This is a new direction of work on which the Department is embarking, with the aim of engaging into multidisciplinary research with the cluster of Biological Sciences at the University of Pretoria.