Faculty of Natural and Agricultural Sciences
School of Mathematical Sciences
Department of Mathematics and Applied Mathematics
Selected Highlights from Research Findings
Deterministic non-linear degenerate parabolic equations have played a crucial role in applied sciences since the 1940's. They model processes as diverse as fluid flows in porous media, gas dynamics and population dynamics.
Most accurate models are expected to take account of randomness.
In this case the arising equations should incorporate some random forcing (white noise, brownian motion, or some of their generalizations). Prof Mamadou Sango's work represents a first attempt at obtaining new existence results for degenerate non-linear stochastic parabolic equations.
The class of equations considered (rather general one) presents some non-linearities that make the usual approach through Galerkin approximation and the method of monotonicity difficult to implement. Sango's result is achieved through the combination of the splitting method (which consists of splitting the equation into a deterministic parabolic equation and a stochastic equation which are both simpler to handle), the monotonicity method and other deep compactness results due to Dubinsky, Prokhorov and Skorokhod.
In a series of papers Prof JMS Lubuma and co-workers produced interesting fundamental results on nonstandard finite difference shemes for the numerical solution of partial differential equations. The basis for this work is the inclusion of additional physical laws in the schemes.
Prof M Sango
Mathematics and Applied Mathematics
+27 (0) 12 842 3679
mamadou.sango@up.ac.za
When shock wave phenomena are studied one is often confronted by the problem of dealing with discontinuous solutions of non-linear partial differential equations. Schwartz distributions work well in this respect when linear differential equations are under consideration, but not for non-linear equations.
A method pioneered by Prof Elemer Rosinger makes it possible to handle jump discontinuities by using order completion to generate so-called Hausdorff continuous solutions. These solutions may have discontinuities in the traditional sense.
Dr R Anguelov
Mathematics and Applied Mathematics
+27 (0) 12 420 2790
roumen.anguelov@up.ac.za
Mathematical problems concerning the motion of incompressible viscous fluids is a major research activity in the department. This involves appropriate boundary conditions for the associated partial differential equations.
Dr Christiaan Le Roux published a paper concerning threshold slip boundary conditions for steady state Stokes flows. It is proved that the differential equations with an appropriate slip boundary condition reminiscent of the behaviour of solids under Coulomb friction represents a well-posed problem.
In the study of boundary permeation in incompressible non-Newtonian fluids, it is often desirable to construct divergence-free solutions of a simple partial differential equation with given boundary behaviour. In a paper by Prof Niko Sauer this is done for the first time by considering in detail the differential geometry of the boundary surface.
Dr C Le Roux
Mathematics and Applied Mathematics
+27 (0) 12 420 2611
christiaan.leroux@up.ac.za
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