Faculty of Natural and Agricultural Sciences
School of Mathematical Sciences
Department of Mathematics and Applied Mathematics
Selected Highlights from Research Findings
The Burgers equation, a simplification of the Navier-Stokes equations, is one of the fundamental model equations in gas dynamics, hydrodynamics and acoustics, which illustrates the coupling between convection/advection. The kinetic energy enjoys boundedness and monotone decreasing properties that are useful in the study of the asymptotic behaviour of the solution. A family of non-standard finite difference schemes, which replicate the energy equality and the properties of the kinetic energy, is constructed. The approach is based on Mickens rule of non-local approximation of non-linear terms. More precisely, the research proposes a systematic non-local way of generating approximations that ensure that the tri-linear form is identically zero for repeated arguments. Numerical experiments are provided that support the theory and demonstrate the power of the non-standard schemes over the classical ones
Contact person: Prof R Anguelov.
A very useful approach to quantum physics is via Feynman path integrals. It provides an alternative to the canonical approach (in terms of a Hamiltonian) to time evolution of quantum dynamical systems. On the other hand, in the context of operator algebras, one can consider so-called non-commutative dynamical systems that provide a mathematical abstraction of quantum dynamical systems and are very convenient for studying structural and conceptual aspects of quantum dynamical systems. These non-commutative dynamical systems are, however, essentially based on the canonical approach. In this research project, an analogue of path integrals, called evolution integrals, was introduced for non-commutative dynamical systems. Evolution integrals are not an abstraction or generalisation of path integrals, but rather an analogous mathematical construction that provides a different point of view on non-commutative dynamical systems, although they are not necessarily of use in physics
Contact person: Dr RdV Duvenhage.
Polymeric liquids exhibit phenomena such as shear-thinning, normal stress differences and pressure-dependence of viscosity, which do not occur in the so-called Newtonian fluids such as water and air. In addition, polymeric liquids do not always satisfy the no-slip boundary condition at solid surfaces. Dr Christiaan le Roux and his co-workers obtained exact solutions for a number of special flow problems for non-Newtonian fluid models with Navier slip boundary conditions. The fluid models considered in these problems were a generalised fluid of complexity, which includes the Navier-Stokes fluid, the power-law fluid and the second-grade fluid as special cases, a Navier-Stokes-like fluid in which the viscosity is a function of the pressure, and a generalisation of the Burgers model for a rate-type viscoelastic fluid. One of the findings is that, in contrast to the Navier-Stokes fluid model, boundary layers can exist even if the flow is sufficiently slow for inertial effects to be negligible. On the other hand, boundary slippage has the effect of diminishing any boundary layers
Contact person: Dr C le Roux.
This work deals with an interesting geometric problem, the deformation of a closed curve into a geodesic on a Finsler manifold. Finsler manifolds originated in the groundbreaking inaugural address of Riemann in 1854 and are characterised by the assignment of a metric that does not only depend on the points on the manifold, but also on the tangent vectors to the manifold. What will later be known as Riemannian manifolds are the simpler case when the metric depends only on the points. After some important work by Finsler, Cartan, Berwald, Chern, Rund and Abkar Zadeh until 1960, the field remained dormant for almost two decades and underwent a resurgence, mainly due to Chern and his students. Certain questions that have found solutions in the Riemannian frame turned out to be extremely difficult to solve in the Finslerian case. One of these problems is that of Harmonic maps from a Finsler manifold into a genuine Finsler manifold. Successful results so far concern the case of a Riemannian target manifold or by endowing a target Finsler manifold with a Sasaki metric, which gives a Riemannian structure to the manifold. The current work treats the case of a one-dimensional Finslerian source manifold as a genuine target Finsler manifold. The study uses the celebrated Eells-Sampson heat flow method
Contact person: Prof M Sango.
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