On a topic chosen by the student from a list of proposals. The project entails progress reports, a written report, an oral presentation and the preparation of a conference poster. Students meet with the coordinator from time to time to be taught generic skills and to discuss progress on the projects. Honneursstudente kies aan die begin van die jaar 'n onderwerp en werk selfstandig, onder leiding van sy/haar projekadviseur, aan die gekose probleem. Aan die einde van die jaar word 'n verslag ingehandig en 'n kort mondelinge voordrag gelewer. Die module behels ook dat studente generiese navorsingsvaardighede aanleer.

General numerical methods for solving flow equations; finite difference/volume methods; procedures for the simulation of diffusive and convective processes; boundary values; solving algorithms such as the SIMPLE range; introduction to CFX. Algemene numeriese metodes vir die oplos van vloeivergelykings; eindige-verskil/volume metodes; prosedures vir simulasie van diffusiewe en konvektiewe prosesse; randwaardes; algoritmes soos die SIMPLE-reeks; inleiding tot CFX.

Focus on numerical methods for matrix computations. Effective solution of square linear systems, least squares problems, the eigenvalue problem. Direct and iterative methods, special attention to sparse matrices and structured matrices. Numerical instability and ill-conditioning. Model problems from partial differential equations and image processing. Fokus op numeriese metodes vir matriksbewerkings. Effektiewe oplos van vierkantige lineêre stelsels, kleinste-kwadrate probleme, die eiewaarde probleem. Direkte en iteratiewe metodes, klem op yl matrikse en matrikse met struktuur. Numeriese onstabiliteit en sleg-geaardheid. Modelprobleme uit parsiële differensiaalvergelykings en beeldverwerking.

Most modelling problems lead to mathematical equations that cannot be solved explicitly. The only recourse in this case is to solve numerically, or to use analytical methods to generate approximate solutions. The latter family of methods forms the focal point of this module. The approximate solution of transcendental equations and differential equations will be discussed, as well as the asymptotic evaluation of integrals that depend on a large or small parameter. Applications include nonlinear oscillators in mechanics, boundary-layer problems in fluids, and a derivation of Stirling's approximation to the factorial function. Numerical methods will be used as a check on the accuracy of the analytical methods. Die meeste modelleringsprobleme lei na wiskundige vergelykings wat nie eksplisiet opgelos kan word nie. Die enigste toevlug in so 'n geval is om numeries op te los, of om analitiese metodes te gebruik om benaderde oplossings te konstrueer. Die metodes in laasgenoemde klas vorm die fokuspunt van hierdie module. Benaderde oplossings tot transendentale vergelykings asook differensiaalvergelykings sal bespreek word, asook die asimptotiese afskatting van integrale met 'n groot of klein parameter. Toepasssings sluit in nie-lineêre ossilators in meganika, randlaagprobleme in vloeistowwe, en 'n herleiding van Stirling se benadering tot die fakulteit-funksie. Numeriese metodes sal gebruik word as bevestiging van die akkuraatheid van die analitiese metodes.

Differential and integral calculus of volume averages in two phase media and its use in the mathematical modelling of transport processes in porous media; the rectangular unit cell model. Differensiaal- en integraalrekening van volumegemiddeldes in twee-fasige media; die modellering van vloeiprossesse in poreuse media; die reghoekige eenheidselmodel.

This course aims to present a general introduction to continuum mechanics. Topics include tensors; kinematics of continuous media; balance of mass, linear and angular momentum, and energy; stress; constitutive theory; linear elasticity; ideal fluids and Newtonian fluids. Die doel van hierdie module is om 'n algemene bekenstelling aan kontinuum meganika te gee. Die onderwerp onder bespreking sluit in tensors: kinematika van kontinuum media: massa- en lineêre en hoeksmomentumbehoud, energiebehoud; spanning, materiaalmodelleringsvergelykings, lineêre elastisiteit, ideale vloeiers, Newtoniese vloeiers.

Following up from AM244, this module goes deeper into the use of Ordinary Differential Equations and Difference Equations for modelling problems in the applied sciences. Most of such problems need to be described with nonlinear terms, making the dynamic behaviour of these models quite varied and sometimes counter-intuitive. For example, dynamics can converge to stationary points (equilibria), but also periodic or quasi-periodic (limit cycles and tori), and deterministically chaotic orbits (strange attractors). Different initial conditions of the system can lead to different types of such asymptotic behaviours. Additionally, the number and type of these attractors can change with model parameters through bifurcations. Most examples of applications will be presented in the field of biology (ecology, evolution, epidemiology), but also in environmental (exploitation of natural resources such as fish stocks) and social sciences (love dynamics)

Broad introduction to graph theory. Problems such as enumeration of graphs; optimal paths in networks; optimal spanning trees; centres and medians; planarity; vertex and edge colouring; Eulerian graphs and Hamiltonicity; tournaments; domination and independence; and Ramsey theory. Breë inleiding tot grafiekteorie. Probleme soos enumerasie van grafieke; optimale paaie in netwerke; optimale spanbome; senters en mediane; planariteit; punt- en lynkleuring; Euler-grafieke en Hamilton-grafieke; toernooie; dominasie en onafhanklikheid; en Ramsey-teorie.

Basic grey-scale transformations and image enhancement techniques in the spatial domain; Fourier analysis in two dimensions and image enhancement techniques in the Fourier domain; image restoration; morphological filters; image compression techniques; image segmentation, representation, description and recognition. Basiese grysskaal-transformasies en beeldverbeteringstegnieke in die fisiese ruimte; Fourier-analise in twee dimensies en beeldverbeteringstegnieke in die Fourier-ruimte; die herstel van beelde; morfologiese filters; tegnieke vir beeldkompaktering; beeldsegmentasie, -voorstelling, -beskrywing en herkenning.

Image interpolation; feature detection and matching; the SVD with application in face recognition; projective geometry; homogeneous coordinates; intersection of parallel lines and the line at infinity; homographies; removing perspective distortion; model estimation with RANSAC; camera models; epipolar geometry; the fundamental matrix; image rectification; the essential matrix; 3D reconstruction.
Beeldinterpolasie; kenmerkdeteksie en -passing; die SVD met toepassing in gesigsherkenning; projeksie-meetkunde; homogene koördinate; snyding van parallelle lyne en die lyn by oneindig; homografieë; verwydering van perspeksie-distorsie; modelafskatting met RANSAC; kamera modelle; epipolêre meetkunde; die fundamentele matriks; beeldregstelling; die essensiële matriks; 3D rekonstruksie.

This course aims to develop an advanced understanding of nonlinear material behaviour. Topics in nonlinear mechanics; nonlinear elasticity; behaviour of elastic-plastic solids, visco-elasticity and non-Newtonian fluids are included. Die doel van hierdie module is om 'n gevorderde verstaan van nie-lineêre materiaal gedrag te ontwikkel. Die onderwerpe onder bespreking sluit in: nie-lineêre meganika; nie-linêre elastisiteit; elasto-plastiese strukture; visco-elastisiteit; en nie-Newtoniese vloeiers.

Derivation of simple partial differential equations (PDEs) from first principles, Fourier analysis, separation of variables and transform techniques for linear second-order PDEs, characteristics, Lagrange's method for first-order PDEs, finite differences. Herleiding van eenvoudige parsiële differensiaalvergelykings (PDVs) uit eerste beginsels, Fourier analise, skeiding van veranderlikes en transform-tegnieke vir lineêre tweede-orde PDVs, karakteristieke, Lagrange se metode vir eerste-orde PDVs, eindige verskille.

Focus on numerical methods for matrix computations. Effective solution of square linear systems, least squares problems, the eigenvalue problem. Direct and iterative methods, special attention to sparse matrices and structured matrices. Numerical instability and ill-conditioning. Model problems from partial differential equations and image processing. Fokus op numeriese metodes vir matriksbewerkings. Effektiewe oplos van vierkantige lineêre stelsels, kleinste-kwadrate probleme, die eiewaarde probleem. Direkte en iteratiewe metodes, klem op yl matrikse en matrikse met struktuur. Numeriese onstabiliteit en sleg-geaardheid. Modelprobleme uit parsiële differensiaalvergelykings en beeldverwerking.