Philosophiae Doctor - PhD (Mathematics)http://hdl.handle.net/11394/32102021-04-13T05:35:20Z2021-04-13T05:35:20ZMeasurements of edge uncolourability in cubic graphsAllie, Imranhttp://hdl.handle.net/11394/78692021-02-25T00:01:45Z2020-01-01T00:00:00ZMeasurements of edge uncolourability in cubic graphs
Allie, Imran
The history of the pursuit of uncolourable cubic graphs dates back more than a century.
This pursuit has evolved from the slow discovery of individual uncolourable
cubic graphs such as the famous Petersen graph and the Blanusa snarks, to discovering
in nite classes of uncolourable cubic graphs such as the Louphekine and
Goldberg snarks, to investigating parameters which measure the uncolourability of
cubic graphs. These parameters include resistance, oddness and weak oddness,
ow
resistance, among others. In this thesis, we consider current ideas and problems regarding
the uncolourability of cubic graphs, centering around these parameters. We
introduce new ideas regarding the structural complexity of these graphs in question.
In particular, we consider their 3-critical subgraphs, speci cally in relation to resistance.
We further introduce new parameters which measure the uncolourability of
cubic graphs, speci cally relating to their 3-critical subgraphs and various types of
cubic graph reductions. This is also done with a view to identifying further problems
of interest. This thesis also presents solutions and partial solutions to long-standing
open conjectures relating in particular to oddness, weak oddness and resistance.
Philosophiae Doctor - PhD
2020-01-01T00:00:00ZFractional black-scholes equations and their robust numerical simulationsNuugulu, Samuel Megamenohttp://hdl.handle.net/11394/76122020-12-03T00:01:31Z2020-01-01T00:00:00ZFractional black-scholes equations and their robust numerical simulations
Nuugulu, Samuel Megameno
Conventional partial differential equations under the classical Black-Scholes approach
have been extensively explored over the past few decades in solving option
pricing problems. However, the underlying Efficient Market Hypothesis (EMH) of
classical economic theory neglects the effects of memory in asset return series, though
memory has long been observed in a number financial data. With advancements in
computational methodologies, it has now become possible to model different real life
physical phenomenons using complex approaches such as, fractional differential equations
(FDEs). Fractional models are generalised models which based on literature have
been found appropriate for explaining memory effects observed in a number of financial
markets including the stock market. The use of fractional model has thus recently
taken over the context of academic literatures and debates on financial modelling.
Philosophiae Doctor - PhD
2020-01-01T00:00:00ZMathematical modeling of TB disease dynamics in a crowded population.Maku Vyambwera, Sibaliwehttp://hdl.handle.net/11394/73572020-10-13T00:00:33Z2020-01-01T00:00:00ZMathematical modeling of TB disease dynamics in a crowded population.
Maku Vyambwera, Sibaliwe
Tuberculosis is a bacterial infection which is a major cause of death worldwide. TB is a
curable disease, however the bacterium can become resistant to the first line treatment
against the disease. This leads to a disease called drug resistant TB that is difficult
and expensive to treat. It is well-known that TB disease thrives in communities in overcrowded
environments with poor ventilation, weak nutrition, inadequate or inaccessible
medical care, etc, such as in some prisons or some refugee camps. In particular, the World
Health Organization discovered that a number of prisoners come from socio-economic disadvantaged
population where the burden of TB disease may be already high and access
to medical care may be limited. In this dissertation we propose compartmental models of
systems of differential equations to describe the population dynamics of TB disease under
conditions of crowding. Such models can be used to make quantitative projections of TB
prevalence and to measure the effect of interventions. Indeed we apply these models to
specific regions and for specific purposes. The models are more widely applicable, however
in this dissertation we calibrate and apply the models to prison populations.
Philosophiae Doctor - PhD
2020-01-01T00:00:00ZEfficient Variable Mesh Techniques to solve Interior Layer ProblemsMbayi, Charles K.http://hdl.handle.net/11394/73242020-09-09T00:00:56Z2020-01-01T00:00:00ZEfficient Variable Mesh Techniques to solve Interior Layer Problems
Mbayi, Charles K.
Singularly perturbed problems have been studied extensively over the past
few years from different perspectives. The recent research has focussed on the
problems whose solutions possess interior layers. These interior layers appear
in the interior of the domain, location of which is difficult to determine a-priori
and hence making it difficult to investigate these problems analytically. This
explains the need for approximation methods to gain some insight into the behaviour
of the solution of such problems. Keeping this in mind, in this thesis
we would like to explore a special class of numerical methods, namely, fitted
finite difference methods to determine reliable solutions. As far as the fitted
finite difference methods are concerned, they are grouped into two categories:
fitted mesh finite difference methods (FMFDMs) and the fitted operator finite
difference methods (FOFDMs). The aim of this thesis is to focus on the
former. To this end, we note that FMFDMs have extensively been used for
singularly perturbed two-point boundary value problems (TPBVPs) whose
solutions possess boundary layers. However, they are not fully explored for
problems whose solutions have interior layers. Hence, in this thesis, we intend firstly to design robust FMFDMs for singularly perturbed TPBVPs whose solutions
possess interior layers and to improve accuracy of these approximation
methods via methods like Richardson extrapolation. Then we extend these
two ideas to solve such singularly perturbed TPBVPs with variable diffusion
coefficients. The overall approach is further extended to parabolic singularly
perturbed problems having constant as well as variable diffusion coefficients.
Philosophiae Doctor - PhD
2020-01-01T00:00:00Z