A bivariate compact finite difference method for the solution of time-dependent PDEs

Project supervisor: Dr Dlamini

Several researchers have used compact finite difference schemes (CFDS) to solve partial differential equations arising in various fields. Normally, the CFDS are used to discretize the space variables and a different technique, usually a time-stepping method, is used for time. However, these techniques limit the accuracy of the difference scheme to fourth order or less in time which means smal mesh sizes must be used in order to get desirable accuracy, and thus much more computational work is involved. Numerical experiments show that implementing the CFDS on both space and time yield highly accurate results. The aim of this work is to apply this idea on nonlinear boundary layer problems.

A predator-prey model with disease in both species

Project supervisor: Dr Chirove

The effect of disease in ecological system is an important issue from mathematical as well as ecological point of view. A predator prey model is proposed incorporating a disease in both species. The objective of the project is determine the effects of a disease introduced to the usual predator prey dynamics. The dynamics are modelled using a simple system on nonlinear ordinary differential equations with nonnegative initial conditions. Threshold conditions in the form of an ecological threshold number will be established from the predator-prey ecosystem to determine the conditions for extinction of species. The disease threshold in form of the reproduction number will be computed to determine whether or not the disease establishes. The basic mathematical analysis will be done using equilibrium, local and global analyses and bifurcation theory. Experimental data simulations and discussion of results will be presented to conclude the study.

Preliminary knowledge required: Calculus, Linear Algebra, Ordinary Differential Equations, Numerical Analysis

A simple compartmental model for gas exchange in insects

Project supervisor: Dr Simelane

Fluid flow in insects can be modeled by systems of partial differential equations (PDEs) describing the axial convection and diffusion of respiratory gases (oxygen and carbon dioxide), as well as exchange between compartments. The modelling is ismilar to the modelling of tracer transport in plants [1], modelling of chromatographic processes [3] and to tracer kinetic modelling in MRI [4]. Buhler et al [2] details a framework for model selection and parameter estimation by inverse modelling. For parameter estimation, inverse modelling requires a nonlinear-paramter optimization in order to least-square minimize the difference of computer simulations and measured experimental data. This technique then requires derivatives of the model parameters in the defined space and time domain. The derivatives are approximated with differential qoutients that are computed in parallel. The described framework also requires forward simulations to perfom in a fast way, while keeping the numerical accuracy above a required minimum. Forward simulation involves the spatial discretization, time discretization of the PDEs and solving the resulting system of nonlinear algebraic equations

The main purpose of this study is to develop a compartmental model that is used for inverse modeling of microscale fluid transport in insects. The developed model will then be solved numerically using techniques acquired in the numerical analysis course. Different spatial discretization methods based on finite difference schemes will be used. To solve the resulting equations, we will apply a Crank-Nicholson solver for the time discretization. The solver will be implemented with fixed time steps in order to not distort the differences between analyzed spatial discretization schemes.

References

1. J. Buhler, G. Huber, E. von Lieres, Finite volume schemes for the numerical simulation of tracer transport in plants, Math. BioSci. 288 (2017) 14-20
2. J. Buhler, E. von Lieres, G. Huber, A class of compartmental models for long-distance tracer-transport in plants, J. Theor. Biol. 341 (2014) 131-142
3. S. Javeed, S. Qamar, A Scidel-Mergenstern, G. Warnecke, Efficient and accurate numerical simulation of nonlinear chromatographic processes, Comput. Chem. Eng. 35 (2011) 2294-2305
4. S.P. Sourbron, D.L. Buckley, Tracer kinetic modelling in MRI: estimating perfusion and capillary permeability Phys. Med. Biol. 57 (2012) R1-R33

An insect inspired pumping model for a viscous flow in a microchannel

Project supervisor: Dr Simelane

Understanding the efficient transport of small amounts of fluids within the insect trachea due to contractions observed in the insect's respiratory system is an important and challenging problem. Its solution can have many applications in science and engineering. For instance, microfluidic devices and micro-electromechanical systems (MEMS) inspired by these physiological systems in insects have in recent years started to gain attention. Microfluidic devices, MEMS and their environments are typically of the order of microns. Microfluidic devices are normally composed of multiple tiny branches that usually move, mix or separate fluids in drug delivery, cell manipulations, DNA chips and ink-jet printer-heads. These applications are usually carried out via transporting microliter amounts of fluids in a controlled manner from one site to another. Two main techniques are currently used to transport fluids at the microscale, namely, conventional pressure-driven flow and valveless mechanisms such as elctro-osmotic, peristaltic and impedance-mismatch. However, these have some functional drawbacks and he current microfabrication techniques, it is difficult to manufacture an entire elastic microfluidic device. Many systems and processes from life sciences, such as the insect respiratory system, can then be imitated in the design of such improved microdevices.

The purpose of the study is to use the observed rhythmic tracheal compressions in insects, which are assumed to be a naturally efficient pumping mechanism in a complex network of microscale channels, to develop a model for microfluidic fluid flow. The student will be expected to develop a model for pumping, controlling and transporting fluids in the insects tracheal network of channels at the microscale. We will explore the Navier-Stokes equations for incompressible flow to present a theoretical model of fluid transport within a channel at the microscale accompanied by wall contractions. We will then investigate the possibility of using these rhythmic contractions as a pumping mechanism. Results of this study can assist in designing and improving valveless microfluidic pumping devices (which might be very useful to mechanical engineers).

References

1. Duncan, F.D., and Bync, M.J., 2002, Respiratory airflow in a wingless beetle, Exper. Biol. 205, pp 2489-2497
2. Simelane, S.M., Abelman, S., and Duncan, F.D., 2017, Microscale gaseous slip flow in the insect trachea and tracheoles, Acta Biotheoretica 65(3), pp 211-231
3. Aboelkassem, Y., and Staples, A.E., 2013, Selective pumping in a network: insect-style microscale flow transport, Bioinspir. Biomim. 8, 22 pp

Allee effects in crime dynamics

Project supervisor: Prof Nyabadza

Allee effects are broadly defined as a decline in individual fitness at low population size or density, that can result in critical population thresholds below which populations crash to extinction. As such, they are very relevant to many ecological s. There are a variety of mechanisms that can create Allee effects, including mating systems, predation, environmental modification, and social interactions among others. The abrupt and unpredicted collapses of many exploited populations is just one illustration of the need to bring Allee effects to the forefront of conservation and management strategies in ecology.

Much of what is known about Allee effects comes from mathematical models. This project looks at the possibility of applying this important ecological concept in reverse to model crime dynamics.

The model considers a population of criminals, N(t), subjected to some “harvesting” by some policing effort E(t). The question we want to answer is at what level of policing effort can we have the total number of criminals crash to extinction?

References

1. Mark Knot, Elements of Mathematical Ecology, 2001
2. Franck Courchamp, Ludek Berec, and Joanna Gascoigne, Allee Effects in Ecology and Conservation, 2008

Dynamics under quartic potentials

Project supervisor: Prof Villet

A quartic potential is one which contains fourth-order terms in the coordinates of the system. This project will require the student to do a literature study of existing publications, identify some simple systems related to those already studied and to perform analytical as well as numerical calculations of the dynamics of these systems.

References

1. W.-H. Steeb, J.A. Louw, C.M. Villet, Maximal one-dimensional Lyapunov exponent and singular-point analysis for a quartic Hamiltonian, Phys. Rev. A, 34, No. 1, 1986
2. W-H Steeb, CM Villet, A Kunick, Chaotic behaviour of a Hamiltonian with a quartic potential, J. Phys. A: Math. Gen., 18 (1985), 2369-3273

Entanglement as a resource

Project supervisor: Dr Kemp

A maximally entangled bipartite state could be considered as a resource for many types of quantum computational tasks, for example quantum teleportation. Given m copies of some partially entangled bipartite state, how many copies n of the maximally entangled state can be obtained just by using local operations and classical communication?

Modelling misdiagnosis in HIV/AIDS infection dynamics

Project supervisor: Prof Nyabadza

The global impact of the scale-up of HIV testing and treatment has been impressive. In 2015, approximately 60% of people with HIV worldwide were aware of their status [1]. As a result by the end of 2015, 17 million people with HIV were on treatment, and global treatment coverage reached 46%. Data from a systematic review of 64 studies (most studies identified were conducted in Africa and other resource-limited settings) are included in this special issue and summarize the magnitude of misdiagnosis in these contexts. The review suggests that on average 0.4% (interquartile range (IQR): 0–3.9%) of diagnoses primarily among adults are false negative and 3.1% (IQR: 0.4–5.2%) are false positive [18]. Among people diagnosed with HIV who were enrolled in care and/or on ART, between 0.1% and 6.6% of patients were reported to be truly HIV negative and had been misdiagnosed [18]. The diagnostic errors identified were largely related to human error [18]. Although reported levels of misdiagnosis are low, if current estimates are accurate [18,19], the large volume of tests conducted each year - over 150 million tests in low- and middle-income countries in 2014 alone, 3 million of which were HIV positive [19] - could result in the misdiagnosis of up to 93,000 people per year if left unaddressed. This work involves designing a model for HIV/AIDS that incorporates the aspect of misdiagnosis. The model is a deterministic model in which a proportion of individuals are misdiagnosed. The formulated differential equations will then be analysed and numerical simulations performed.

References

1. UNAIDS. Global AIDS update. Geneva: Joint United Nations Programme on HIV/AIDS; 2016
2. Johnson C, Fonner V, Sands A, Ford N, Obermeyer C, Tsui S, et al. To err is human, to correct is public health: identifying poor quality testing and misdiagnosis of HIV status, J Int AIDS Soc. 2017; 20(Suppl 6):21755.
3. WHO. Factsheet to the WHO consolidated guidelines on HIV testing services. Geneva: World Health Organization;2015.

Modelling and dynamics of a coupled pendulum

Project supervisor: Mr Anderson

We consider two mathematical simple pendulums with the masses attached to each other via a spring. We shall derive the equations of motion, solve the resulting differential equations and simulate and study long-term dynamics.

Preliminary knowledge required: Multivariable Calculus, Linear Algebra. (Knowledge of Hamiltonian Mechanics will be beneficial.)

Modelling within host dynamics for Langerhans HIV infection with immune system harvesting

Project supervisor: Dr Chirove

Langerhans cells are antigen-presenting immune cells foreskin, vaginal, and oral mucosa of humans are the initial cellular targets in the sexual transmission of HIV-1 and a target, reservoir, and vector of dissemination. Their role can be a major pathogenic process in HIV-1 infection and development of acquired immunodeficiency syndrome (AIDS). A model is proposed to capture the interaction of HIV and Langerhans infections within the host. The immune system response against HIV is represented by processes analogous to the ecological constant yield harvesting and constant effort harvesting. We ask, what is the critical harvesting strategy by the immune system that can destroy the virus in finite time? The dynamics of Langerhans cells infection with HIV will be captured using a system of nonlinear ordinary differential equations where basic thresholds such as the basic reproduction number will be computed. Simple steady state analysis and stability will be done using the basic stability theory and numerical simulations will be carried in MATHEMATICA or Octave check the predictive capacity of the mathematical models with experimental data. Results will be discussed and validated against the biological processes involved the interactions.

Preliminary knowledge required: Calculus, Linear Algebra, Ordinary Differential Equations, Numerical Analysis. (Knowledge on the biology of the interaction is not a pre-requisite and is easy to gain through course of the study. You do not need to have done any biology for you to do this project)

Numerical simulations in special relativistic hydrodynamics

Project supervisor: Dr Herbst

Numerical simulations of gas dynamics in astrophysics remains of paramount importance for the understanding of various aspects of stellar and galaxy evolution. In the most extreme cases (such as relativistic gas jets from black holes) special relativity is required to accurately simulate the underlying dynamics. In the project, the student will learn the basics behind constructing a numerical scheme for the relativistic hydrodynamic equations. Emphasis will be placed on solvers for the primitive variables which are not provided directly from the numerical simulations.

Polynomials of the bifurcations points of some maps

Project supervisor: Prof Villet

The bifurcation points of the logistic map are algebraic and have been and the polynomials satisfying these points have been investigated in good mathematical detail. This project will consist of an analytical study of similar polynomials for maps which are topologically conjugate to the logistic map.

References

1. J Blackhurst, Polynomials of the bifurcation points of the logistic map, Int. J. of Bifurcation and Chaos, 21, No. 7, (2011)

Simulating photon trajectories past a rotating black hole

Project supervisor: Dr Herbst

Since the first direct observation of the M87 black hole there has been increased interest in the dynamics of photons around strong gravitational sources. The aim of this project is to introduce the student to two concepts involved in the simulation of such observations:

1. the general relativistic principles governing the motion of photons in the vicinity of a black hole, and
2. the robust numerics required to accurately simulate these governing equations.

Coding, which will be performed in Mathematica, is a central part of the project with the focus on being the computational component rather than the physics component.

Simulating stomata to mitigate climate change

Project supervisor: Prof Momoniat

Stomata are tiny openings or pores on the surface of leaves that facilitate the intake of carbon dioxide and limite water loss from the leaf. Stomata open and closes as a result of the diffusion process that takes place at the leaf surface. In this project we will model the diffusion process describing the functioning of stomata. Artifical intelligence tools will be used to determine the paraemters in the model. The main aim of the project is to determine ways in which information obtained from A.I can be used to stimulate stomata and minimise the effects of climate change.

References

1. V. Chandra, K. Pushkar, Topic on Botany: Anatomical feature in relation to taxonomy, Competition Science Vision, (2005), 795-796.
2. R. J. Ferry, Stomata, Subsidiary Cells, and Implications, MIOS Journal, 9 (2008), 9-16.

Space-time bivariate spectral collocation method for parabolic PDEs

Project supervisor: Dr Dlamini

Spectral collocation methods are powerful numerical techniques used to solve differential equations. They have historically been used to approximate boundary value problems and the spatial part of a PDE. Generally, a low-order finite difference approximation of the time derivative is used for the temporal variable. This is not ideal since in most cases the time discretization error overcomes the spatial discretization error. The application of spectral methods on both time and space variables have gained significant attention recently. In this work, the space-time spectral method will be used to solve parabolic PDEs. Numerical examples will be presented to show that this formulation has expontential rates of convergence in both space and time.

Wave breaking on smooth/rough approximation to a beach

Project supervisor: Prof Momoniat

In this project the student will model the breaking of water waves on a smooth/rough shore. The shallow water equations and the Boussinesq equations are used to model the flow of the waves up the iclined shore. The conditions for wave-breaking are determined in terms of the physical paremeters.

References

1. Ge Wei, J. T. Kirby, S. T. Grilli, R, Subramanya, A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves, J. Fluid Mech. 294 (1995) 71-92.

The stabilizer formalism and quantum error correction

Project supervisor: Dr Kemp

The stabilizer formalism is a formalism that encompasses many aspects of quantum operations. It aims to study the set of transformations that leave a quantum state invariant. These operations form part of what is known as the Pauli group. We will review this formalism and its application to quantum error correction.

Topological Data Analysis

Project supervisor: Dr Visaya

Topological data analysis (TDA) is an emerging field in Applied Mathematics whose goal is to provide mathematical and algorithmic tools in understanding the topological and geometric structure of (possibly high-dimensional or complex) data. Although still rapidly evolving, TDA now provides a set of mature and efficient tools that can be used in combination or complementary to other data sciences tools. Depending on the interests and strengths of the student, the honours project will discuss TDA in relation dynamical systems or machine learning, and can be applied to any type of data (e.g. financial, medical, chaotic, or biological data).