Programme & Abstracts

SANUM 2009

The 33rd AnnualSouth African Symposium on

Numerical and Applied Mathematics

6, 7 and 8 April

Hosted by theDepartment of Mathematical Sciences

Stellenbosch UniversitySouth Africa

SANUM is sponsored by the International Council for Industrial and Applied Mathematics (ICIAM) under their support scheme for developing countries.

Welcome to SANUM 2009!

Registration takes place on Monday 6 April from 08:15 to 08:50 in the foyer of the general engineering building, located on the corner of Banhoek and Joubert roads (numbered “1” on the map).

Internet Access

Wired network access is available in room A308.

● Automatic proxy configuration:

http://dip.sun.ac.za/local/proxy.pac

Login: sanumPassword: sanum2009

● Outgoing mail (SMTP) server:

mail.sun.ac.za

Lunch is served at the Botanical Gardens at 13:00 daily. This is also the venue for the welcoming reception on Monday at 18:30, where we meet around light snacks and drinks.

The conference dinner is held on the Eaglevlei Wine Estate, 10 minutes' drive outside Stellenbosch. We leave from the parking lot in front of the engineering building at 18:00.

On the cover: Panoramic view of Stellenbosch from Papegaaiberg (Afrikaans for Parrot Hill). Photo by Francois Malan.

Time Monday 6 April Tuesday 7 April Wednesday 8 April

08:15—09:00 Registration

09:00—09:45 Plenary IMaini

Mathematical modelling of some aspects of solid tumour growth

Plenary IPeirce

Hydraulic Fractures: multiscale phenomena, asymptotic and numerical solutions

Plenary IFornberg

Radial Basis Functions for Solving Partial Differential Equations

09:45—10:30 Plenary IIH.T. Banks

Propagation of Uncertainty in Dynamical Systems

Plenary IIRamage

Adaptive Grid Methods for Q-Tensor Theory of Liquid Crystals

Plenary IIFlyer

Applications of Radial Basis Functions in the Geosciences

10:30—11:00 Tea

11:00—11:45 Plenary IIIKappel

Modeling the Impact of BMI on Toxin Concentrations in Hemodialysis Patients

Plenary IIIJoly

Asymptotic Models in Aeroacoustics

Plenary IIIMilewski

Solitary non-integrable wave-packets: stability and dynamics

11:45—12:15 AcklehDeterministic and

stochastic juvenile-adult models with

application to amphibians

TokmanConstruction and performance of

exponential integrators

ChabassierEnergy preserving scheme for a nonlinear system of piano strings

PatidarOn fitted numerical methods for delay

differential equations

MasonTurbulent fluid fracture

BandaA Multi-level Dynamic

Compressor Optimization Approach for Gas Flow

Networks

12:15—12:45 J. BanksReliability of the use of surrogate species in risk assessment: a

population matrix model approach

D. LaurieRational

approximations to the complex error

function

KaraSymmetry invariance of

conservation laws

MunyakaziOn the convergence

acceleration techniques for singularly perturbed turning point problems

MakindeOn Thermal Stability of a Non-Newtonian Reactive Flow in a Cylindrical Pipe with Convective cooling

NgnotchouyeNumerical

Implementation of One Dimentional Systems of Conservation Laws on

Networks

13:00—14:15 Lunch (Botanical Gardens)

14:15—14:45

Mathem

atics in Biology

Special S

ession on

SuttonUncertainty and

Estimation in Cell Proliferation

Modeling

AbelmanNumerical Treatment

of First-kind Abel Integral Equations

Vision and Learning

Special S

ession on

De PierroThe Generalized

Attenuated Radon Transform and Its

Inversion

BashierAn efficient numerical

method for a delay differential model of two

cooperating species

ChollomStable Hybrid Methods for

the Solution of Stiff Systems of Ordinary Differential

Equations

Van der BijlRefinement preservation

in the bivariate case

14:45—15:15 SimonsThe Volcano

Effect in Bacterial Chemotaxis

PrenticeStability regions of RKGL and related

methods

Fabris-RotelliClustering by LULU

Scale-Space Filtering

AhmedOn control strategies for co-infection of TB and HIV via mathematical

modelling

15:15—15:45 H. LaurieA two-patch model for savanna

plant/herbivore interaction, with an application to

grazing lawns

AnguelovStructurally Stable

Numerical Schemes for Applied

Dynamical Models

HerbstProbabilistic PCA

SuleimanMathematical analysis to determine some control

strategies for a co-infection model of HIV

and malaria

15:45—16:15 Tea

16:15—16:45 YusufA mathematical

model for controlling the

spread of the HIV/AIDS pandemic

Van RensburgSolvability of a model for the

vibration of a beam with a damping tip

body

HoffmannActive appearance

models fitting

KhabirReproducing Kernel Element Method for

solving an option pricing models

16:45—17:15 FarbodMoment

Estimation for Some Discrete Distributions

Generated by Stable Laws

Van der MerweNumerical simulation of the vibration of a Timoshenko beam

with boundary damping

ColvilleIntroduction to the

Centre for High Performance Computing

SidahmedUse of mesh free

methods to price an American option

18:30 Welcoming Reception(Botanical Gardens)

Conference Dinner(Eaglevlei Wine Estate)

5

Monday 06 April

MONDAY PLENARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Mathematical modelling of some aspects of solid tumour growthProf. Philip Maini, University of Oxford . . . . . . . . . . . . . . . . . . . . . . . . 6

Propagation of Uncertainty in Dynamical SystemsProf. H.T. Banks, N.C. State University . . . . . . . . . . . . . . . . . . . . . . . . 6

Modeling the Impact of BMI on Toxin Concentrations in Hemodialysis PatientsProf. Franz Kappel, University of Graz . . . . . . . . . . . . . . . . . . . . . . . . 6

SPECIAL SESSION ON MATHEMATICAL BIOLOGY . . . . . . . . . . . . . . . . 6

Deterministic and stochastic juvenile-adult models with application to amphibiansProf. Azmy Ackleh, University of Louisiana at Lafayette . . . . . . . . . . . . . . . 6

Reliability of the use of surrogate species in risk assessment: a population matrix modelapproachDr John E. Banks, University of Washington . . . . . . . . . . . . . . . . . . . . . 6

Uncertainty and Estimation in Cell Proliferation ModelingDr Karyn Sutton, North Carolina State University . . . . . . . . . . . . . . . . . . 7

The Volcano E�ect in Bacterial ChemotaxisJulie Simons, University of Wisconsin, Madison . . . . . . . . . . . . . . . . . . . 7

A two-patch model for savanna plant/herbivore interaction, with an application to grazinglawnsHenri Laurie, University of Cape Town . . . . . . . . . . . . . . . . . . . . . . . . 7

A mathematical model for controlling the spread of the HIV/AIDS pandemicTajudeen Yusuf, University of the Western Cape . . . . . . . . . . . . . . . . . . . 7

Moment Estimation for Some Discrete Distributions Generated by Stable LawsDavood Farbod, Yerevan State University . . . . . . . . . . . . . . . . . . . . . . . 7

MONDAY MORNING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Construction and performance of exponential integratorsProf. Mayya Tokman, University of California, Merced . . . . . . . . . . . . . . . 8

Rational approximations to the complex error functionProf. Dirk Laurie, Stellenbosch University . . . . . . . . . . . . . . . . . . . . . . . 8

MONDAY AFTERNOON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Numerical Treatment of First-kind Abel Integral EquationsProf. Shirley Abelman, University of the Witwatersrand . . . . . . . . . . . . . . . 8

Stability regions of RKGL and related methodsDr Justin Prentice, University of Johannesburg . . . . . . . . . . . . . . . . . . . . 8

Structurally Stable Numerical Schemes for Applied Dynamical ModelsProf. Roumen Anguelov, University of Pretoria . . . . . . . . . . . . . . . . . . . . 8

Solvability of a model for the vibration of a beam with a damping tip bodyProf. Nic van Rensburg, University of Pretoria . . . . . . . . . . . . . . . . . . . . 9

Numerical simulation of the vibration of a Timoshenko beam with boundary dampingDr Alna van der Merwe, CPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6 Monday 06 April

MATHEMATICAL MODELLING OFSOME ASPECTS OF SOLID TU-MOUR GROWTH

Prof. Philip Maini (University of Oxford)

We use a number of mathematical modelling tech-niques (ODEs, PDEs, hybrid cellular automata andother discrete cell-based approaches) to address anumber of problems in cancer biology. Applicationswill include escape from homeostatis in colorectalcancer, somatic evolution in breast cancer and mul-tiscale modelling in vascular tumour growth.

PROPAGATION OF UNCERTAINTYIN DYNAMICAL SYSTEMS

Prof. H.T. Banks (N.C. State University)

After discussing the shortcomings of a crypto-deterministic formulation, we compare two other ap-proaches for inclusion of uncertainty/variability inmodeling growth in size-structured population mod-els. One entails imposing a probabilistic structureon growth rates in the population that leads to ran-dom di�erential equations while the other involvesformulating growth as a stochastic Markov di�usionprocess, resulting in forward Kolmogorov or Fokker-Planck equations. We present a theoretical analysisthat allows one to include comparable levels of un-certainty in the two distinct formulations in makingcomparisons of the two approaches. Computationalaspects of the approaches are also discussed.

MODELING THE IMPACT OF BMION TOXIN CONCENTRATIONS INHEMODIALYSIS PATIENTS

Prof. Franz Kappel (University of Graz)

Recent studies have shown that dialysis patients witha higher BMI on the long run do better than pa-tients with a low BMI provided they undergo thesame treatment. One explanation is that a largervolume of adipose tissue has a bu�ering function, sothat the toxin concentration in extracellular �uid islower during the interdialytic periods. We present acompartment model which is capable to model thise�ect. Furthermore, we discuss problems associatedwith parameter estimation for this model.

DETERMINISTIC AND STOCHAS-TIC JUVENILE-ADULT MODELSWITH APPLICATION TO AMPHIB-IANS

Prof. Azmy Ackleh (University of Louisiana atLafayette)

We develop a deterministic juvenile-adult modelwhere juveniles are structured by age and adults arestructured by size. This model is described by asystem of nonlinear �rst order hyperbolic equations.Existence-uniqueness of weak solutions is discussedand convergence of a �nite di�erence approximationto the unique weak solutions is established. This �-nite di�erence approximation is then used to com-pare the model output, via a least-squares approach,to statistical population estimates of green tree frogs,which are obtained from capture-mark-recapture �elddata. To test the e�ect of demographic stochasticity,we develop stochastic versions of the deterministicmodel which account for randomness in birth death,and age/size changes and present numerical resultsfor the dynamics of these stochastic models.

RELIABILITY OF THE USE OFSURROGATE SPECIES IN RISK AS-SESSMENT: A POPULATION MA-TRIX MODEL APPROACH

Dr John E. Banks (University of Washington)

The use of surrogate species is an important toolfor predicting the e�ects of management decisionsor the establishment of protective measures for en-dangered/threatened species. However, relying on ahandful of common species as a means of predict-ing the fate of a wide suite of endangered species ispotentially fraught with problems, especially whenstatic metrics such as LD/LC 50 are used. Using amatrix model, we derive a closed-form mathemati-cal expression aimed at determining when surrogatespecies population outcomes will reliably predict out-comes of target (listed) species. In particular, wedevelop an inequality that indicates how dissimilarlife history traits (survival & fecundity) of the sur-rogate and listed species may be before the surro-gate species outcomes indicate a positive populationgrowth, whereas the listed species is driven to extinc-tion. We compare our derived criterion to several �shsurrogate species and their target salmonid species,and discuss more general applications of our �ndings.

UNCERTAINTY AND ESTIMATIONIN CELL PROLIFERATION MODEL-ING

Dr Karyn Sutton (North Carolina State University)

Changes in cell proliferation are indicative of diseaseprogression or the worsening of a medical conditionsin many instances, with cancer being an obvious ex-ample. The standardization of �uorescent labelingof cells as measured by �ow cytometry has allowed

Monday Morning 7

for potential in quanti�cation of proliferation dynam-ics. We discuss a model of cell proliferation dynamicswhere the �uorescent label is a structure variable. Weoutline areas in which uncertainty should be explic-itly incorporated in the model and discuss the estima-tion of parameters via inverse problem methodology.

THE VOLCANO EFFECT IN BAC-TERIAL CHEMOTAXIS

Julie Simons (University of Wisconsin, Madison)

A classic problem in biology is to model organ-isms at the population level, circumventing thecomplexity of describing each individual's behavior.Population-level models are either written outrightby postulating reasonable macroscopic behavior (of-ten as �ux laws with sources/sinks), or by describ-ing the phenomena on an organism scale and de-ducing these laws. In bacterial chemotaxis�the de-scription of biased motion of bacteria towards opti-mal environments�the simplest macroscopic modelsare written with Keller-Segel (KS) type equations.Such models are able to capture many aspects of thebehavior of a population, and only recently have beendeduced from a reasonable organism-level descriptionof behavior. There are, however, situations wherethe bacterial behavior is not captured by KS models:we focus in particular on experimental and computa-tional results which demonstrate that under certainconditions bacteria will aggregate some distance awayfrom a peak in its attractant. In this work, we pro-pose a microscopic explanation for this phenomenonand present population- level models that go beyondthe KS models, derived from organism-level behavior.

A TWO-PATCH MODEL FOR SA-VANNA PLANT/HERBIVORE IN-TERACTION, WITH AN APPLICA-TION TO GRAZING LAWNS

Henri Laurie (University of Cape Town)

We formulate a two-patch version of Norman Owen-Smith's GMM model for biomass density of a ide-alised herbivore feeding on a idealised resource in aseasonal environment. The formulation is a straight-forward extension: GMM on each patch and mi-gration between patches. Patch areas are explicit.We assume that migration may be in�uenced by theamount of vegetation in the local patch, but is inde-pendent of other factors. The seasonality introducestime-dependence in the parameters, so that the re-sulting equations are not autonomous. We brie�yreview some analytical results for the analogical au-tonomous system and display some computational re-sults for the full system. We then apply the model

to a system consisting of two types of vegetation, onewith low carrying capacity but fast growth and highernutrition which we claim mimics the role of grazinglawns. The results appear to indicate that grazinglawns are disadvantageous to animal growth underGMM.

A MATHEMATICAL MODEL FORCONTROLLING THE SPREAD OFTHE HIV/AIDS PANDEMIC

Tajudeen Yusuf (University of the Western Cape)

Authors: T.T.Yusuf and F.Benyah We present adeterministic model for the spread and control ofHIV/AIDS infection. The model is used to examinevarious combinations of control e�orts in reducing thespread of the pandemic.

MOMENT ESTIMATION FOR SOMEDISCRETE DISTRIBUTIONS GEN-ERATED BY STABLE LAWS

Davood Farbod (Yerevan State University)

Davood Farbod Yerevan State University In bioin-formatics there are well-known discrete distribu-tions that have properties like Stable Laws. Weare constructing in such discrete distributions. Inthe present article an algorithm which is based onNewton-Raphson method to moment estimator of theindex α for some truncated discrete distributions gen-erated by maximally skewed Stable Laws is proposed.Numerical examples are studied and support our the-ory. Taking into account that the logarithm transfor-mation of such discrete random variable has �nitemoments of any order, we consider moment estima-tion for logarithm of it. Key words: Index α; MomentEstimate; Newton-Raphson Method; Stable Laws.

CONSTRUCTION AND PERFOR-MANCE OF EXPONENTIAL INTE-GRATORS

Prof. Mayya Tokman (University of California,Merced)

Exponential integrators o�er an alternative way tointegrate nonlinear systems of ODEs compared to ex-plicit and implicit methods. A particularly advanta-geous characteristic of exponential schemes is their ef-�ciency in integrating large sti� systems. While �rstexponential schemes were introduced in early 1960'sthey haven't attained wide popularity since they wereconsidered prohibitively expensive. A proposal tocombine Krylov projection algorithm and exponentialintegration alleviated this computational constraint

8 Monday 06 April

and was followed by a resurgence of interest in thesemethods. In this talk we will describe what are thebuilding blocks in construction of an e�cient expo-nential integrator and provide an overview of researche�orts in development of competitive schemes of thistype. We will discuss such important aspects of thesemethods as optimization of computational complex-ity per integration step, derivation of order conditionsusing B-series and development of adaptive integra-tion strategies. Performance of exponential integra-tors will be evaluated using a suite of test problemsand comparing them with standard methods.

RATIONAL APPROXIMATIONS TOTHE COMPLEX ERROR FUNC-TION

Prof. Dirk Laurie (Stellenbosch University)

We start from Weideman's 1994 paper on evalua-tion of the complex error function (also known asthe Faddeeva function) by a formula valid in all ofthe upper half-plane. That formula can be viewedas an (n− 1, n) rational approximation in which thedenominator has a single pole of appropriate multi-plicity, and can be derived by transforming a certainauxiliary function to the upper half-plane to the unitdisk by a Möbius transformation that maps the polein question to in�nity, followed by Taylor expansionaround the origin. Instead of Taylor expansion, weuse near-best rational approximation on the unit cir-cle to obtain the same accuracy with n reduced by afactor of more than two.

NUMERICAL TREATMENT OFFIRST-KIND ABEL INTEGRALEQUATIONS

Prof. Shirley Abelman (University of the Witwater-srand)

A rational basis set is constructed using [1; 1] ratio-nal interpolation. The basis is used in a product in-tegration method for solving �rst-kind Abel integralequations.

STABILITY REGIONS OF RKGLAND RELATED METHODS

Dr Justin Prentice (University of Johannesburg)

We describe the RKrGLm method for solving ODEs,and we show the stability regions of RK5GL3, whichis a fully explicit realization of RKrGLm. We con-sider implicit forms of the method in which theGL component is replaced with closed Newton-Cotes(NC) quadrature, and we suggest that if the un-derlying Runge-Kutta (RK) method is explicit, the

method will likely never be A-stable. If RK is im-plicit, however, the possibility of A-stability does ex-ist. We show that the method IRK3NC4 seems to beA(α)-stable, with α ≈ 89◦.

STRUCTURALLY STABLE NUMER-ICAL SCHEMES FOR APPLIED DY-NAMICAL MODELS

Prof. Roumen Anguelov (University of Pretoria)

We consider the construction of reliable numericaldiscretizations of continuous dynamical systems aris-ing as models of di�erent natural phenomena, with afocus on schemes which correctly replicate the prop-erties of the original dynamical systems. The work isbased on the new concept of topological dynamic con-sistency, which describes in precise terms the align-ment of the properties of the discrete dynamical sys-tem and the approximated continuous dynamical sys-tem. This development is based on the importantobservation that the "dynamics" of a dynamical sys-tem, e.g. �xed points, periodic orbits, non-wanderingset, and the way these �nal states are approached, aretopological properties. Consequently, we consider thesimilarity of a continuous dynamical system with theassociated numerical methods, viewed as discrete dy-namical systems, in the sense of topological equiva-lence between the corresponding evolution operators.In this conceptual setting the structural stability ofthe evolution operators of the original system and itsdiscretization play an important role. We recall thatthe notion of structural stability is de�ned with re-spect to certain functional space, typically the spaceof all smooth maps. For models in practical applica-tions this space is often not applicable and we showhow one can design an appropriate space V for a spe-ci�c model, so that the evolution operator is V stable.Our approach is to construct a discretization suchthat the evolution operator of the respective discretedynamical system is in V and is also V structurallystable. Then the topological dynamic consistency ofthe schemes follows in a natural way. In the construc-tion of the schemes we use Mickens' non-standard �-nite di�erence method. A numerical procedure forthe classic endemic model for the spread of disease isderived by using nonlocal approximation of the non-linear term. The topological dynamic consistency ofthe scheme is obtained via the structural stability ofthe evolution operators with respect to a suitably de-�ned function space V.

SOLVABILITY OF A MODEL FORTHE VIBRATION OF A BEAMWITH A DAMPING TIP BODY

Prof. Nic van Rensburg (University of Pretoria)

Monday Afternoon 9

We consider a model for the vibration of a beam witha damping tip body in a recent article (2002). We de-rive a variational form for the motion of the beam anduse it to prove that the model problem has a uniqueweak solution. The proofs are based on existenceresults for a general linear vibration model problempublished in 2002. It is important to realize that theexistence of solutions for problems involving partialdi�erential equations is not a black or white issue;there exists an hierarchy of possible solutions. Wepresent elementary examples to illustrate this fact.

NUMERICAL SIMULATION OFTHE VIBRATION OF A TIMO-SHENKO BEAMWITH BOUNDARYDAMPING

Dr Alna van der Merwe (CPUT)

The Timoshenko beam model consists of two partialdi�erential equations; one for the de�ection of thebeam and another for the angle due to rotation ofa cross section. The e�ect of boundary damping onthe vibration of a Timoshenko beam has been stud-ied previously by solving the associated eigenvalueproblem. For a problem of this nature, spectral anal-ysis may not be su�cient to predict the dynamic be-haviour of the beam. We present examples of nu-merical simulations of the transverse vibration of aTimoshenko beam with boundary damping. In par-ticular, these examples illustrate how the e�ect ofthe boundary damping depends in a signi�cant wayon the initial state of the beam.

10 Monday 06 April

Tuesday 07 April

TUESDAY PLENARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Hydraulic Fractures: multiscale phenomena, asymptotic and numerical solutionsProf. Anthony Peirce, University of British Columbia . . . . . . . . . . . . . . . . 11

Adaptive Grid Methods for Q-Tensor Theory of Liquid CrystalsDr Alison Ramage, University of Strathclyde . . . . . . . . . . . . . . . . . . . . . 11

Asymptotic Models in AeroacousticsDr Patrick Joly, INRIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

TUESDAY MORNING I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Energy preserving scheme for a nonlinear system of piano stringsJuliette Chabassier, INRIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Symmetry invariance of conservation lawsProf. Abdul Hamid Kara, University of the Witwatersrand . . . . . . . . . . . . . . 12

TUESDAY MORNING II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

On �tted numerical methods for delay di�erential equationsProf. Kailash C. Patidar, University of the Western Cape . . . . . . . . . . . . . . 12

On the convergence acceleration techniques for singularly perturbed turning point problemsJustin B. Munyakazi, University of the Western Cape . . . . . . . . . . . . . . . . 12

SPECIAL SESSION ON COMPUTER VISION . . . . . . . . . . . . . . . . . . . . . 12

The Generalized Attenuated Radon Transform and its inversion: application in X-RayFluorescence TomographyProf. Alvaro De Pierro, University of Campinas . . . . . . . . . . . . . . . . . . . 12

Clustering by LULU Scale-Space FilteringInger Fabris-Rotelli, University of Pretoria . . . . . . . . . . . . . . . . . . . . . . 13

Probabilistic PCAProf. Ben Herbst, Stellenbosch University . . . . . . . . . . . . . . . . . . . . . . . 13

Active appearance models �ttingMcElory Ho�mann, University of Stellenbosch . . . . . . . . . . . . . . . . . . . . 13

Introduction to the Centre for High Performance ComputingKevin Colville, CHPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

TUESDAY AFTERNOON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

An e�cient numerical method for a delay di�erential model of two cooperating speciesEihab B. M. Bashier, University of the Western Cape . . . . . . . . . . . . . . . . 13

On control strategies for co-infection of TB and HIV via mathematical modellingHasim A. O. Ahmed, University of the Western Cape . . . . . . . . . . . . . . . . 14

Mathematical analysis to determine some control strategies for a co-infection model of HIVand malariaSara M. A. Suleiman, University of the Western Cape . . . . . . . . . . . . . . . . 14

Reproducing Kernel Element Method for solving an option pricing modelsMohmed H. M. Khabir, University of the Western Cape . . . . . . . . . . . . . . . 14

Use of mesh free methods to price an American optionAbdelmgid O. M. Sidahmed, University of the Western Cape . . . . . . . . . . . . . 14

Tuesday Morning I 11

HYDRAULIC FRACTURES: MUL-TISCALE PHENOMENA, ASYMP-TOTIC AND NUMERICAL SOLU-TIONS

Prof. Anthony Peirce (University of BritishColumbia)

Hydraulic fractures (HF) are a class of tensile frac-tures that propagate in brittle materials by the in-jection of a pressurized viscous �uid. In this talkI introduce models of HF and provide examples ofnatural HF and situations in which HF are used inindustrial problems. Natural examples of HF includethe formation of dykes by the intrusion of pressurizedmagma from deep chambers. They are also used ina multiplicity of engineering applications, including:the deliberate formation of fracture surfaces in gran-ite quarries; waste disposal; remediation of contami-nated soils; cave inducement in mining; and fractur-ing of hydrocarbon bearing rocks in order to enhanceproduction of oil and gas wells. Novel and emergingapplications of this technology include CO2 seques-tration and the enhancement of fracture networks tocapture geothermal energy. I describe the govern-ing equations in 1-2D as well as 2-3D models of HF,which involve a coupled system of degenerate non-linear integro-partial di�erential equations as well asa free boundary. I demonstrate, via re-scaling the1-2D model, how the active physical processes mani-fest themselves in the HF model and show how a bal-ance between the dominant physical processes leadsto special solutions. I discuss the challenges for e�-cient and robust numerical modeling of the 2-3D HFproblem including: the rapid construction of Green'sfunctions for cracks in layered elastic media, robustiterative techniques to solve the extremely sti� cou-pled equations, and a novel Implicit Level Set Algo-rithm (ILSA) to resolve the free boundary problem.The e�cacy of these techniques is demonstrated withnumerical results.

ADAPTIVE GRID METHODS FORQ-TENSOR THEORY OF LIQUIDCRYSTALS

Dr Alison Ramage (University of Strathclyde)

In addition to the defect core structure, the dynam-ics of defect movement is also a crucial issue for liq-uid crystal cells. Previous numerical simulations ofsuch e�ects have mostly involved using standard �-nite di�erence or �nite element techniques on uniformgrids, and have often been based on using directortheory rather than the Q-tensor approach proposedhere. However, the presence in the physical problemof characteristic lengths with large scale di�erences

(the size of the defect is very small compared to thatof the cell) suggests that more sophisticated numer-ical modelling techniques could be used to great ef-fect. One obvious approach is to use an adaptive gridtechnique, ensuring that there is no waste of compu-tational e�ort in areas where there is no need for a �negrid. Adaptive grid methods have been successfullyused to solve PDEs in many branches of computa-tional mathematics such as computational �uid dy-namics, mathematical biology, semiconductor mod-elling and aerospace engineering. In this talk we willpresent an introductory study of the use of adaptivegrid methods for solving PDE problems in Q-tensortheory of liquid crystals. This work is in collaborationwith Dr Christopher J. Newton of Hewlett-PackardLaboratories in Bristol, UK.

ASYMPTOTIC MODELS IN AEROA-COUSTICS

Dr Patrick Joly (INRIA)

In many applications in aeroacoustics, in particularin aeronautics, it is important to simulate the in-teractions of acoustic waves with boundaries (walls,wings,...). Moreover, boundary layers of very smallwidth are present close to the boundaries. Thequestion of a satisfactory treatment of boundaries inaeroacoustics remains essentially open. In this work,we study acoustic wave propagation in a thin duct,in the presence of a laminated �ow, a problem whichcan be seen as a �rst step towards the treatment ofsuch boundary layers. We use an asymptotic analy-sis when the width of the duct is small with respectto the wavelength, which amounts to a low frequencyanalysis, assuming that the Mach pro�le of the �ow isobtained by a scaling of a reference pro�le M(y). Weestablish a limit model and study its well-posedness.On the one hand, we exhibit some pro�les for whichthe model is strongly ill-posed and obtain, as a by-product, new instability results for compressible lin-earized Euler equations. On the other hand, we es-tablish su�cient conditions on M(y) for the modelto be well posed : the weak well-posedness of theCauchy problem is obtained via a quasi-explicit rep-resentation of the solution, using the Fourier-Laplacetransform in space-time. Numerical illustrations willbe presented. Finally, we shall show how these resultscan be used to construct e�ective boundary condi-tions for boundary layers.

ENERGY PRESERVING SCHEMEFOR A NONLINEAR SYSTEM OFPIANO STRINGS

Juliette Chabassier (INRIA)

12 Tuesday 07 April

The linear wave equation does not describe the com-plexity of the piano strings vibration enough forphysics based sound synthesis. The nonlinear cou-pling between transversal and longitudinal modes hasto be taken into account, as does the �geometricallyexact� model. This system of equations can be clas-si�ed among a general energy preserving class of sys-tems. We present an implicit, centered, second orderaccurate, numerical scheme that preserves a discreteenergy, leading to unconditional stability of the nu-merical scheme. The complete model takes into ac-count the bridge coupling the strings, and the ham-mer non linear attack on the strings.

SYMMETRY INVARIANCE OFCONSERVATION LAWS

Prof. Abdul Hamid Kara (University of the Witwa-tersrand)

The use of symmetry properties of a given systemof partial di�erential equations to generate new con-servation laws from known conservation laws and theconcept of basis of conservation laws has been investi-gated. Furthermore, it has been shown that one canconstruct new conservation laws for a given systemof di�erential equations by the action of any symme-try generators on the known conservation laws of thegiven system of di�erential equations without havingto transform the symmetry generators to canonicalsymmetry generators. The generated conservationlaws are non-trivial if the given system of di�eren-tial equations are derivable from a variational prin-ciple. The role of `multipliers' has been thoroughlyinvestigated - particularly, in the construction of new/higher-order conservation laws. There is a determin-ing system for �nding ALL multipliers (and hence allconservation laws) for any given PDE system. Thedetermining system consists of the adjoint of the sym-metry determining equation plus certain extra de-termining equations (called adjoint-invariance condi-tions) formulated entirely on the solution space. Thiscomes from applying the Euler operator to the multi-plier expression and then splitting the resulting equa-tions.

ON FITTED NUMERICAL METH-ODS FOR DELAY DIFFERENTIALEQUATIONS

Prof. Kailash C. Patidar (University of the WesternCape)

Several numerical methods have been developed inthe past to solve complex di�erential models. How-ever, it is di�cult to extend some or most of thesemethods to solve the delay di�erential equations

(DDEs). In this talk, we will consider some DDEsand describe construction and analysis of some �ttednumerical methods for such problems.

ON THE CONVERGENCE ACCEL-ERATION TECHNIQUES FOR SIN-GULARLY PERTURBED TURNINGPOINT PROBLEMS

Justin B. Munyakazi (University of the WesternCape)

Convergence acceleration techniques have been dis-cussed a lot for singularly perturbed problems with-out turning point problems. However, very few worksare found the turning point problems. In this talk,we will consider a singularly perturbed turning pointproblem and �rstly discuss construction, analysis andimplementation of some novel numerical methods.Then we discuss about how we accelerate the con-vergence of these methods.

THE GENERALIZED ATTENU-ATED RADON TRANSFORM ANDITS INVERSION: APPLICATION INX-RAY FLUORESCENCE TOMOG-RAPHY

Prof. Alvaro De Pierro (University of Campinas)

X-ray Computed Tomography (CT) is one of themost important imaging techniques for medical di-agnosis. Mathematically, it consists of inverting thewell known Radon Transform, that is nothing butthe set of line integrals of the function we want to re-construct. In other applications, like Single PhotonEmission Computed Tomography (SPECT), wherethe function to be reconstructed is photon emissioninside the body, an attenuation factor appears un-der the integral, giving rise to a more di�cult inver-sion. In X-ray Fluorescence Computed Tomography(XFCT), the emission is produced by the excitationby an outer source of intense X-rays. This gener-ates a still more complicated operator containing theattenuation of the X-rays and of the emitted �uores-cence as well. In this work, we describe the analyticand the iterative inversion of the XFCT Transform.We deduce new mathematical formulas using ideasfrom recent transforms that appear in the solution ofBoundary Value Problems. We also present experi-mental results with simulated as well as synchrotrongenerated real data.

CLUSTERING BY LULU SCALE-SPACE FILTERING

Inger Fabris-Rotelli (University of Pretoria)

Tuesday Afternoon 13

Multidimensional LULU operator theory has beenderived and the application of it in Image Processinginvestigated - speci�cally, their application to clus-tering of images and object detection. Their advan-tage over standard clustering is due to the knowledgeof the image structures at all scales, hence the Scale-Space theory, as well as the intrinsic shape propertiesof objects at all scales which is all available throughthe decomposition of the image by the LULU opera-tors, the Discrete Pulse Transform.

PROBABILISTIC PCA

Prof. Ben Herbst (Stellenbosch University)

I'll give an overview of probabilistic PCA, and illus-trate it with examples.

ACTIVE APPEARANCE MODELSFITTING

McElory Ho�mann (University of Stellenbosch)

Active appearance models (AAMs) (Cootes et al.,1998; Edwards et al., 1998) are deformable templatemodels using shape and texture (grayscale or colourinformation) to segment objects of interest from animage. They are a generalisation of active shapemodels (ASMs) (Cootes et al., 1995) that only useshape information. In this talk, I shall discuss �ttingmethods of AAMs with an emphasis on non- lineartechniques. References Cootes, T. F., Edwards, G. J.,Taylor, C. J., 1998. Active appearance models. In:European Conference on Computer Vision. Vol. 2.pp. 484�498. Edwards, G. J., Taylor, C. J., Cootes,T. F., 1998. Interpreting face images using active ap-pearance models. In: FG '98: Proceedings of the 3rd.International Conference on Face & Gesture Recog-nition. IEEE Computer Society, Washington, DC,USA, pp. 300�305. Cootes, T., Taylor, C., Cooper,D., Graham, J., 1995. Active Shape Models-TheirTraining and Application. Computer Vision and Im-age Understanding 61 (1), 38�59.

INTRODUCTION TO THE CENTREFOR HIGH PERFORMANCE COM-PUTING

Kevin Colville (CHPC)

The Centre for High Performance Computing(CHPC) is a multi- disciplinary national resourceproviding high performance computer resources forresearch and development. These resources includea 640 processor distributed cluster with 1.25 TB ofRAM, a 32 processor shared memory machine, and91 TB in a high speed storage array network (SAN).

The CHPC is also hosts the 4096 processor IBM BlueGene/P of the IBM Blue Gene for Africa project. TheCHPC aims to enhance signi�cant research, addressgrand challenges, and grow computational researchinto a viable mode alongside experiment and theoryacross all academic disciplines. The CHPC's commit-ment to the advancement of science through the facil-itation of world-class science is apparent in its promo-tion and facilitation of the use of computational tech-nologies and research techniques; fostering of innova-tion through e�ective partnerships; and training of anew generation of computationally-skilled researchersin areas underpinned by high-end computing. TheCentre for High Performance Computing (CHPC) isan initiative of the Department of Science and Tech-nology and is managed by the Meraka Institute of theCSIR.

AN EFFICIENT NUMERICALMETHOD FOR A DELAY DIF-FERENTIAL MODEL OF TWOCOOPERATING SPECIES

Eihab B. M. Bashier (University of the WesternCape)

Authors: E.B.M. Bashier and K.C. Patidar Abstract:We consider a system of two coupled partial delay dif-ferential equations (PDDEs) describing the dynam-ics of two cooperative species. The original systemis reduced to a system of ordinary delay di�erentialequations (DDEs) obtained by applying the methodof lines. Then we construct a �tted operator �nitedi�erence method (FOFDM) to solve this resultingsystem. The model considered in this paper is verysensitive to small changes in the parameters associ-ated in with the model. Depending on the values ofthese parameters, the solution can be stable, peri-odic and/or aperiodic. Such behavior of the solutionis exploited via the proposed FOFDM. This FOFDMis analyzed for convergence and it is seen that thismethod is unconditionally stable and has the accu-racy of O(k + h2), where k and h denote time andspace step-sizes, respectively. Some numerical re-sults con�rming theoretical observations are also pre-sented. These results are comparable with those ob-tained in the literature.

ON CONTROL STRATEGIES FORCO-INFECTION OF TB ANDHIV VIA MATHEMATICAL MOD-ELLING

Hasim A. O. Ahmed (University of the WesternCape)

In this talk we consider a co-infection model of TB

14 Tuesday 07 April

and HIV. By going through the sub-cases, we will dis-cuss the derivation of the optimal system. Then wediscuss appropriate numerical methods to solve theresulting system.

MATHEMATICAL ANALYSISTO DETERMINE SOME CON-TROL STRATEGIES FOR A CO-INFECTION MODEL OF HIV ANDMALARIA

Sara M. A. Suleiman (University of the WesternCape)

In this talk, we will discuss a mathematical model ofHIV and Malaria co-infection. Through some quali-tative analysis, the derivation of an optimal controlmodel will then be discussed.

REPRODUCING KERNEL ELE-MENT METHOD FOR SOLVING ANOPTION PRICING MODELS

Mohmed H. M. Khabir (University of the WesternCape)

We will review some option pricing models using thedi�erential equation approach and then we will dis-cuss the application of the Reproducing Kernel El-ement Method to solve a di�erential model to pricethe American put options.

USE OF MESH FREE METHODS TOPRICE AN AMERICAN OPTION

Abdelmgid O. M. Sidahmed (University of the West-ern Cape)

Mesh free methods are used a lot to solve the par-tial di�erential equation models in the past. In thistalk, we will describe how to use these approachesto price an American option using the Black-Scholesframework.

Tuesday 07 April 15

Wednesday 08 April

WEDNESDAY PLENARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Radial Basis Functions for Solving Partial Di�erential EquationsProf. Bengt Fornberg, University of Colorado . . . . . . . . . . . . . . . . . . . . . 16

Applications of Radial Basis Functions in the GeosciencesDr Natasha Flyer, NCAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Solitary non-integrable wave-packets: stability and dynamicsProf. Paul Milewski, University of Wisconsin . . . . . . . . . . . . . . . . . . . . . 16

WEDNESDAY MORNING I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Turbulent �uid fractureProf. David Mason, University of the Witwatersrand . . . . . . . . . . . . . . . . . 16

On Thermal Stability of a Non-Newtonian Reactive Flow in a Cylindrical Pipe with Con-vective coolingProf. Oluwole Daniel Makinde, Cape-Peninsula University of Technology . . . . . 17

Stable Hybrid Methods for the Solution of Sti� Systems of Ordinary Di�erential EquationsDr Joshua Chollom, University of Jos . . . . . . . . . . . . . . . . . . . . . . . . . 17

WEDNESDAY MORNING II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

A Multi-level Dynamic Compressor Optimization Approach for Gas Flow NetworksProf. Mapundi Banda, University of the Witwatersrand . . . . . . . . . . . . . . . 17

Numerical Implementation of One Dimentional Systems of Conservation Laws on NetworksJ. Medard T. Ngnotchouye, University of KwaZulu-Natal . . . . . . . . . . . . . . 17

Re�nement preservation in the bivariate caseRinske van der Bijl, Stellenbosch University . . . . . . . . . . . . . . . . . . . . . . 18

16 Wednesday 08 April

RADIAL BASIS FUNCTIONS FORSOLVING PARTIAL DIFFEREN-TIAL EQUATIONS

Prof. Bengt Fornberg (University of Colorado)

For the task of solving PDEs, �nite di�erence (FD)methods are particularly easy to implement. Finiteelement (FE) methods are more �exible geometri-cally, but tend to be di�cult to make very accurate.Pseudospectral (PS) methods can be seen as a limitof FD methods if one keeps on increasing their orderof accuracy. They are extremely e�ective in many sit-uations, but this strength comes at the price of verysevere geometric restrictions. A more standard intro-duction to PS methods (rather than via FD meth-ods of increasing orders of accuracy) is in terms ofexpansions in orthogonal functions (such as Fourier,Chebyshev, etc.). Radial basis functions (RBFs) were�rst proposed around 1970 as a tool for interpolatingscattered data. Since then, both our knowledge aboutRBFs and their range of applications have growntremendously. After introducing them, we will focuson their use for solving a wide range of PDEs (whichincludes PDEs in the geosciences, as Dr. NatashaFlyer will describe in a following presentation). Inthis context of solving PDEs, we can see the RBFapproach as a major generalization of PS methods,abandoning the orthogonality of the basis functionsand in return obtaining vastly improved simplicityand �exibility. Spectral accuracy becomes now eas-ily available also when using completely unstructuredmeshes, permitting local node re�nements in criticalareas. A very counterintuitive parameter range (mak-ing all the RBFs very �at) turns out to be of specialinterest. Computational cost and numerical stabilitywere initially seen as potential di�culties, but majorprogress have recently been made also in these areas.

APPLICATIONS OF RADIAL BA-SIS FUNCTIONS IN THE GEO-SCIENCES

Dr Natasha Flyer (NCAR)

Current community models in the geosciences employa variety of numerical methods from �nite-di�erence,�nite-volume, �nite- or spectral-elements, to pseu-dospectral methods. All have specialized strengthsbut also serious weaknesses. The �rst three meth-ods are generally considered low-order and can in-volve high algorithmic complexity (as in triangularelements or unstructured meshes). Global spectralmethods do not practically allow for local mesh re-�nement and often involve cumbersome algebra, es-pecially in spherical geometry. Radial basis functionshave the advantage of being spectrally accurate for

arbitrary node layouts in multi-dimensions with ex-treme algorithmic simplicity, and naturally permitlocal node re�nement. We will show test examplesranging from vortex roll-ups, modeling idealized cy-clogenesis, to 2-D unsteady nonlinear �ows posed bythe shallow water equations to 3-D mantle convec-tion in the earth's interior. Their performance willbe evaluated based on numerical accuracy, stabilityand computational performance.

SOLITARY NON-INTEGRABLEWAVE-PACKETS: STABILITY ANDDYNAMICS

Prof. Paul Milewski (University of Wisconsin)

Often, the dynamics of slowly modulated wavepack-ets are studied with envelope type equations such asthe Nonlinear Schrödinger Equation and its relatives.In some situations, however, there can be serious is-sues with such a modelling assumption, and we dis-cuss an alternative: a "primitive" equation (govern-ing the waves themselves not their envelope) whichappears to be a more realistic model to the stabilityand dynamics of such waves.

TURBULENT FLUID FRACTURE

Prof. David Mason (University of the Witwater-srand)

The propagation of a two-dimensional turbulent �uidfracture in impermeable rock is considered. The �uid�ow is turbulent if the Reynolds number is su�cientlyhigh.The PKN approximation is made in which themean �uid pressure in the fracture is proportional tothe fracture half-width.The partial di�erential equa-tion for the half- width has the form of of a nonlin-ear di�usion equation with a parameter which takesvalues depending on whether the �ow is laminar orturbulent. A similarity solution is derived for a pre-existing one- sided turbulent �uid fracture by consid-ering a linear combination of the Lie point symme-tries of the nonlinear di�usion equation. The e�ectof turbulence on the propagation of the fracture isinvestigated.

ON THERMAL STABILITY OF ANON-NEWTONIAN REACTIVEFLOW IN A CYLINDRICAL PIPEWITH CONVECTIVE COOLING

Prof. Oluwole Daniel Makinde (Cape-Peninsula Uni-versity of Technology)

A large class of non-Newtonian �uids used in in-dustries is chemically reactive (e.g. coal slurries,

Wednesday Morning II 17

polymer solutions or melts, drilling mud, hydrocar-bon oils, grease) when subjected to external heatingand high shear. This may lead to a high temper-ature being generated within the �uid. Hence, thestudy of heat transfer and thermal criticality of re-active non-Newtonian �uids is extremely importantin order to ensure safety of life and properties dur-ing handling and processing of such �uids. In thispaper, the e�ect of convective cooling on a reactivethird-grade �uid �owing steadily through a cylindri-cal pipe is investigated. It is assumed that the sys-tem exchange heat with the ambient following New-ton's cooling law and the reaction is exothermic un-der Arrhenius kinetics, neglecting the consumptionof the material. The simpli�ed governing nonlinearequations of momentum and energy are obtained andsolved using a special type of the Hermite�Padé ap-proximation technique. Important properties of theoverall �ow structure including velocity �eld, tem-perature �eld, bifurcations, and thermal criticalityconditions are obtained and discussed quantitatively.Keywords: Cylindrical pipe; Third-grade �uid; Ar-rhenius kinetics; Thermal criticality, Hermite�Padéapproximants,Convective cooling. Reference: [1] Y.Tourigny, P.G. Drazin, The asymptotic behaviour ofalgebraic approximants, Proc. Roy. Soc. Lond.A456 (2000) 1117�1137. [2] O. D. Makinde. Anal-ysis of non- Newtonian reactive �ow in a cylindricalpipe. Journal of Applied Mechanics, Vol. 76, 034502(1-5), 2009. [3] O. D. Makinde. On the Chebyshevcollocation spectral approach to stability of �uid �owin a porous medium. Int. J. Numer. Meth. Fluids59:791�799, 2009. [4] O. D. Makinde. Thermal crit-icality for a reactive gravity driven thin �lm �ow ofa third-grade �uid with adiabatic free surface downan inclined plane. Appl. Math. Mech. -Engl. Ed.Vol 30(3), 373-380, 2009. [5] O. D. Makinde. Ther-mal stability of a reactive viscous �ow through aporous-saturated channel with convective boundaryconditions. Applied Thermal Engineering, Vol. 29,1773�1777, 2009.

STABLE HYBRID METHODS FORTHE SOLUTION OF STIFF SYS-TEMS OF ORDINARY DIFFEREN-TIAL EQUATIONS

Dr Joshua Chollom (University of Jos)

The construction of A stable Block Hybrid AdamsMoulton Formulae (BH) for the solution of sti� sys-tems of Ordinary Di�erential Equations of order k+2is discussed. These methods are constructed fork=2,3 by incorporating one o� grid point for eachstep number.The methods are self-starting and haveshown to posses attractive stability and convergenceproperties. A numerical experiment carried to inves-

tigate the accuracy of the method reveals its e�ciencywhen compared with the state of the art sti� Mat labODE solver (ode 23).

A MULTI-LEVEL DYNAMIC COM-PRESSOR OPTIMIZATION AP-PROACH FOR GAS FLOW NET-WORKS

Prof. Mapundi Banda (University of the Witwater-srand)

We present an e�cient approach for dynamic com-pressor optimisation in gas networks based on space-mapping. For the �ne space a nonlinear isothermalgas �ow model is applied while as for the coarse spacean algebraic model is employed. To solve an algebraicmodel is cheaper than solving a nonlinear model. Itis desirable though to bring as much nonlinear e�ectsinto the optimisation as possible. The mathemat-ical formulation and algorithmic aspects of such aspace- mapping approach will be presented. Compu-tational results and comparison between di�erent ap-proaches is presented. The results demonstrate thatsuch a multi-level approach provides accurate resultse�ciently.

NUMERICAL IMPLEMENTATIONOF ONE DIMENTIONAL SYSTEMSOF CONSERVATION LAWS ONNETWORKS

J. Medard T. Ngnotchouye (University of KwaZulu-Natal)

In this talk we discuss some techniques for the imple-mentation of a numerical approach for approximatinga system of conservation laws in a network of tubes,rivers, or channels. The dynamic of the vertex is ofmajor interest. The key result here is on the couplingof di�erent models for each arc at the junction. Afterprescribing a generally nonlinear algebraic system ofcoupling conditions at the junction, the theory of LaxRiemann Solvers is used to render the system solv-able. The implementation combines a solver for thenonlinear coupling conditions and a non-oscillatoryhigh-order method for the solution of the �ow equa-tions.

REFINEMENT PRESERVATION INTHE BIVARIATE CASE

Rinske van der Bijl (Stellenbosch University)

We show how to generate re�nable functions fromprevious ones in such a way that re�nability is pre-served. As a foremost example of a re�nable function,

18 Wednesday 08 April

we shall make use of various box splines throughout.For the case where the dilation matrix is twice theidentity matrix, we obtain results for arbitrary direc-tion matrices, whereas restricting our attention to di-agonal dilation matrices in general. Not only is suchre�nement preservation useful for the theory it opensup, but it moreover leads to the enhancement of thesmoothness class of the re�nable function.

Wednesday 08 April 19

Participants

Surname First name(s) A�liation E-mail

Abelman Shirley University of the Witwatersrand Shirley.Abelman@wits.ac.za

Aboiyar Terhemen University of Agriculture, Makurdi taboiyar@hotmail.com

Ackleh Azmy University of Louisiana at Lafayette ackleha@bellsouth.net

Ahmadinia Mahdi Qom University ahmadinia@mail.uk.ac.ir

Ahmed Hasim A. O. University of the Western Cape hahmed@uwc.ac.za

Anguelov Roumen University of Pretoria roumen.anguelov@up.ac.za

Banda Mapundi University of the Witwatersrand mapundi.banda@wits.ac.za

Banks H.T. N.C. State University htbanks@ncsu.edu

Banks John E. University of Washington banksj@u.washington.edu

Bashier Eihab B. M. University of the Western Cape 2547456@uwc.ac.za

Benyah Francis University of the Western Cape fbenyah@uwc.ac.za

Brink Willie University of Stellenbosch willie.brink@gmail.com

Chabassier Juliette INRIA juliette.chabassier@inria.fr

Chikwasha Vasco Harare Institute of Technology vchikwasha@gmail.com

Chollom Joshua University of Jos chollomj@yahoo.com

Colville Kevin CHPC kevin.colville+SANUM@gmail.com

De Pierro Alvaro University of Campinas alvaro@ime.unicamp.br

Fabris-Rotelli Inger University of Pretoria inger.fabris-rotelli@up.ac.za

Farbod Davood Yerevan State University d.farbod@gmail.com

Flyer Natasha NCAR �yer@ucar.edu

Fornberg Bengt University of Colorado fornberg@colorado.edu

Gavhi Rejoyce University of Stellenbosch rejoya@sun.ac.za

Herbst Ben Stellenbosch University herbst@sun.ac.za

Ho�mann McElory University of Stellenbosch mcelory@sun.ac.za

Joly Patrick INRIA patrick.joly@inria.fr

Kappel Franz University of Graz franz.kappel@uni-graz.at

Kara Abdul Hamid University of the Witwatersrand Abdul.Kara@wits.ac.za

Khabir Mohmed H. M. University of the Western Cape 2769908@uwc.ac.za

Laurie Dirk Stellenbosch University dpl@sun.ac.za

Maini Philip University of Oxford maini@maths.ox.ac.uk

Makinde Oluwole Daniel Cape-Peninsula University of Technology makinded@cput.ac.za

Mason David University of the Witwatersrand David.Mason@wits.ac.za

Mavungu Masiala University of Johannesburg masmavp@yahoo.fr

Milewski Paul University of Wisconsin milewski@math.wisc.edu

Muller Neil University of Stellenbosch neil@dip.sun.ac.za

20 Wednesday 08 April

Munyakazi Justin B. University of the Western Cape 2482357@uwc.ac.za

Muthivhi Patrick Mutshutshu thavhakhulu@yahoo.uk

Ogbogbo Chisara Peace University of Ibadan chisaraogbogbo@yahoo.com

Okosun Kazeem University of the Western Cape okoare@uwc.ac.za

Patidar Kailash C. University of the Western Cape kpatidar@uwc.ac.za

Peirce Anthony University of British Columbia peirce@math.ubc.ca

Prentice Justin University of Johannesburg jprentice@uj.ac.za

Ramage Alison University of Strathclyde A.Ramage@strath.ac.uk

Roux Jeanne-marie Stellenbosch University jmsroux@gmail.com

Schlunz Bernard Stellenbosch University 14795612@sun.ac.za

Shrivastava Rajeev mpsidc Rajiv.Srivastav@gmail.com

Sidahmed Abdelmgid O. M. University of the Western Cape 2836781@uwc.ac.za

Simons Julie University of Wisconsin, Madison simons@math.wisc.edu

Suleiman Sara M. A. University of the Western Cape ssuleiman@uwc.ac.za

Sutton Karyn North Carolina State University klsutton@ncsu.edu

Swart Albert Stellenbosch University adswart@gmail.com

T. Ngnotchouye J. Medard University of KwaZulu-Natal 207521833@ukzn.ac.za

Tokman Mayya University of California, Merced mtokman@ucmerced.edu

Van Rensburg Nic University of Pretoria nvrens@scientia.up.ac.za

Van der Bijl Rinske Stellenbosch University rinske@sun.ac.za

Van der Merwe Alna CPUT vandermerwea@cput.ac.za

Weideman André University of Stellenbosch weideman@sun.ac.za

Yusuf Tajudeen University of the Western Cape 2875661@uwc.ac.za

Van der Walt Stéfan Stellenbosch University stefan@sun.ac.za