Faculty of Natural and Agricultural Sciences
School of Mathematical Sciences
Department of Mathematics and Applied Mathematics
Selected Highlights from Research Findings
Singular perturbation problems arise in various fields of science and engineering. The main concern with such problems is the rapid and large variation of their solutions in one or more narrow "layer region(s)" where the solutions jump from one stable state to another or to prescribed boundary values. The singular perturbation problems considered here are mostly dispersive in the sense that the solutions experience a global phenomenon of rapid oscillation throughout the entire interval or domain. Consequently, classical methods always fail in providing reliable numerical results, as the singular perturbation parameter and the mesh size cannot vary independently. To overcome this difficulty, we develop some fitted operator methods. More precisely, using the non-standard finite difference approach, the dispersive (or the dissipative) nature of the solutions is captured in the schemes by systematically constructing suitable denominator functions for the discrete second order derivatives. The research was a collaboration between Prof JM Lubuma from the University and KC Patidar from the University of the Western Cape.
Contact person: Prof JM-S Lubuma.
Quantum Gravity has recently emerged as a major alternative to String Theory in what is by far the most important outstanding problem in modern Physics, namely, finally bringing together General Relativity and Quantum Theory. This synthesis has been in the making, yet so far never achieved, for most of the last century. One of the basic mathematical difficulties involved in such a synthesis is a suitable accommodation of continuous and discontinuous structures. Much of modern Mathematics deals with mostly continuous, and in fact, even more smooth structures, a situation which leads to discontinuities appearing as singularities which are most difficult to deal with. Several important and general consequences in which an unprecedented large class of singularities can be dealt with by rather classical algebraic and topological means, such as encountered in the study of usual Euclidean spaces were recently developed. Specifically, the singularities can form such large sets as to be dense among the non-singular, that is, regular points. In particular, the set of singularities can even be larger than the set of regular or non-singular points. Such structures have earlier been called “space-time foam”. However, prior to the author's work, no rigorous mathematical construction has been given for them. In order to suit the state of the art mathematical apparatus used in Physics, the construction of such space-time foam structures is extended from usual Euclidean spaces to manifolds. In this regard, a remarkable feature of the construction is the considerable ease and generality in dealing with such large classes of singularities on manifolds, when compared to earlier attempts in the literature.
Contact person: Prof EE Rosinger.
Vibration analysis is a major area of research in mechanics. Of particular and growing interest is the study of impact oscillators, which arise in a variety of engineering and real-life problems such as rattling gears, vibrating absorbers, car suspensions and impact print hammers. These oscillators are nonsmooth mechanical systems with complex and even chaotic behaviours. Typically, at the time of the impact, which is supposed to be instantaneous, the velocity of the oscillator changes instantaneously to the extent that this may induce unwanted phenomena such as excessive noise wear and fatigue. Thus, there is an essential need to study these systems properly in order to improve their performance. A recent paper is an extension of the authors' earlier work on vibro-impact problems with a single degree of freedom. The setting of this study varies from multi-dimensional impact oscillators with or without damping to the situation when the impact is a consequence of the coupling of oscillators. For each mechanical model, we have constructed a variety of non-standard finite difference schemes. Apart from their excellent error bounds and unconditional stability, the schemes are analysed for their efficiency to preserve some important physical properties of the systems including, the conservation of energy between consecutive impact times, the periodicity of the motion and the boundedness of the solutions. The research was a collaboration between Prof JM Lubuma and Y Dumont from the Université de la Réunion.
Contact person: Prof JM-S Lubuma.
Special functions play a significant role in mathematical physics, especially in boundary value problems. Usually we call a function special when the function, just as the logarithm, the exponential and trigonometric functions, belongs to the toolbox of the applied mathematician, the physicist or engineer. The function usually has a particular notation and a number of properties of the function are known. Dr Kerstin Jordaan worked with Prof Kathy Driver, dean of the Faculty of Natural Sciences at the University of Cape Town (UCT) and Prof Andrei Martínez Finkelshtein from the University of Almería in Spain on this research project. The hypergeometric function 2F1 was first defined by Gauss in 1812, yet there is still research being done on properties of this special function. By letting one of the parameters be a negative integer, the function reduces to an orthogonal polynomial. It is well-known that the zeros of two adjacent polynomials in an orthogonal sequence separate each other. This means that there is exactly one zero of one polynomial between the two zeros of the next polynomial. Apart from providing more detail on the location of the zeros, separation results are also useful in other contexts, such as interpolation problems. We study separation properties of the zeros of two hypergeometric polynomials in different orthogonal sequences. Similar work is done for the zeros of other classical orthogonal polynomials such as Laguerre, Gegenbauer and Jacobi polynomials. The analysis and behaviour of special functions and their rational approximants are important research questions. A rational approximant is a ratio of two polynomials that approximate the function and where the two polynomials in the approximant satisfy specified conditions. In order to obtain a good rational approximant to a given function, one needs to know the location of the zeros of the polynomial in the denominator of the approximant. Typically, one needs this information in order to avoid those regions where the poles of the approximant lie. The 3F2 polynomial is a generalised hypergeometric 2F1 polynomial. The 2F1 polynomial is orthogonal and this orthogonality immediately yields several properties, most importantly the property that all the zeros are real and restricted to an interval on the real line. The 3F2 polynomials do not have this orthogonality property and as a result, very little is known about the zeros of these polynomials. We use methods from a related field of Mathematics, namely Pólya frequency functions and sequences used in combinatorics, to prove that the zeros of some families of 3F2 hypergeometric polynomials are all real and negative.
Contact person: Dr KH Jordaan.
The analysis of mathematical models describing the interaction of sound and elastic waves in an acoustic chamber with a flexible “wall” has been the subject of extensive research during the past few decades. Interest in the problem of blood flow has led to the consideration of fluid-structure models, i.e. structural acoustic models in which the acoustic medium is a viscous fluid. Recently interest in investigations of the circulatory system has centered on the role of the shear stress which the flowing blood exerts on the artery (wall shear stress) rather than on pressure stress and there is considerable evidence linking wall shear stress with vein permeability. Against this biological background, we consider a fluid-structure model comprising the equations for a Stokes fluid coupled with the equations for the quasi-longitudinal deflections of a two-dimensional plate, augmented by additional terms representing the shear stress which the fluid exerts on the plate. A variational approach is used to show unique solvability of the interactive system of equations while uniform stability of the energy associated with the model is established under a condition which links the potential plate energy with the viscosity of the fluid.
Contact person: Dr M Grobbelaar-van Dalsen.
It was in 1870 that Helmholtz observed that a vector field could be decomposed into a part which is divergence-free and a part which is the gradient of a potential. This observation could, with the advent of Hilbert space theory, be described as a particular orthogonal decomposition of vector fields. The particular decomposition plays a cardinal role in the study of incompressible fluid motion. When the fluid motion is more complex, under the influence of dynamic boundary conditions for example, the so-called Helmholtz-Weyl decomposition cannot be used. In a recent paper a very general theory is developed which takes care of more situations. When applied to specific situations the results are surprising and stunningly beautiful.
Contact person: Prof N Sauer.
We frequently assume that returns are normally distributed for the statistical modelling of financial markets. Normal return based models perform well for small asset return fluctuations; however, markets do exhibit large moves occasionally which are not modelled well by the normal distribution assumption. We consider an application of extreme value theory (EVT) to the FTSE/JSE TOP40 index, especially in the light of recent big moves. We show the difference between returns obtained using the frequent assumption of normal returns and that obtained using EVT. We conclude that EVT is a very useful tool for realistic risk management analyses.
Contact person: Prof E Maré.
Mathematics educators worldwide have identified a serious problem: despite the importance of the mathematical sciences and the opportunities and remuneration available to graduates of mathematics, fewer students are enrolling for degrees in mathematics. Furthermore, many of those who do enrol do not have a clear idea of what professional work as a mathematician entails. The authors of a recent paper are conducting an extended project investigating mathematics students’ perceptions about mathematics and about working as professionals in the mathematical sciences, as well as the impact that these ideas have on their learning of mathematics. Almost 1,200 students in five countries completed a short survey including three open-ended questions asking about their views of mathematics and its role in their future studies and planned professions. Students’ conceptions of mathematics ranged from the narrowest view as a focus on calculations with numbers, through a notion of mathematics as a focus on models or abstract structures, to the broadest view of mathematics as an approach to life and a way of thinking. Broader conceptions of mathematics were more likely to be found in later-year students and there were significant differences between universities. One of the teaching and learning implications from the study are that there is a clear need to emphasize the broader conceptions of mathematics even in the training of engineering students, let alone those who will become professional mathematical scientists or mathematics educators.
Contact person: Prof AF Harding.
Mathematics educators worldwide have identified a serious problem: despite the importance of the mathematical sciences and the opportunities and remuneration available to graduates of mathematics, fewer students are enrolling for degrees in mathematics. Furthermore, many of those who do enrol do not have a clear idea of what professional work as a mathematician entails. The authors of a recent paper are conducting an extended project investigating mathematics students’ perceptions about mathematics and about working as professionals in the mathematical sciences, as well as the impact that these ideas have on their learning of mathematics. Almost 1,200 students in five countries completed a short survey including three open-ended questions asking about their views of mathematics and its role in their future studies and planned professions. Students’ conceptions of mathematics ranged from the narrowest view as a focus on calculations with numbers, through a notion of mathematics as a focus on models or abstract structures, to the broadest view of mathematics as an approach to life and a way of thinking. Broader conceptions of mathematics were more likely to be found in later-year students and there were significant differences between universities. One of the teaching and learning implications from the study are that there is a clear need to emphasize the broader conceptions of mathematics even in the training of engineering students, let alone those who will become professional mathematical scientists or mathematics educators.
Contact person: Prof JC Engelbrecht.
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