Theses
This collection contains the Doctoral and Masters by Research theses produced within the department.
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Analysis of a ReactionDiffusion Model Towards Description and Prediction of Breast Cancer ProgressionThis thesis conducts a thorough investigation by combining mathematical analysis, empirical clinical data, and meticulous clinical observations to investigate the tumour dynamics which are controlled by two crucial biomarkers: cancerous cells (CK7) and active Tcells (CD4). Introducing a reactiondiffusion model facilitates the spatial and temporal distribution of the biomarkers CK7 and CD4. Subsequently, explore the stability, bifurcation and sensitivity analyses to show the effects of critical biological parameters on tumour dynamics. An exploration of the travelling wave solutions of the model, through numerical methods, aids in the visualisation of the dynamic behaviour of the system, particularly the invasion of the CK7 population towards the CD4 population. Employing a numerical approach, specifically utilising the finite volume method, provides a clear representation of the dynamic interplay within the tumour environment. Our analysis contributes not only to a heightened theoretical understanding but also holds potential implications for therapeutic decisions in tumour dynamics. Hence, our investigation shows efficacy not only in predicting tumour dynamics but also in providing a robust framework to present specific clinical observations.

Numerical Methods for Stochastic AllenCahn Equation and Stochastic Subdiffusion and SuperdiffusionIn this Thesis, we consider the numerical solution of stochastic partial differential equations with particular interest on the Ɛdependent AllenCahn equation, and the stochastic time fractional partial differential equations in both subdiffusion and superdiffusion cases.

Social Sustainability in the Oil and Gas Industry in the Niger Delta RegionSocial sustainability is a critical component of the Sustainable Development Goals and net zero targets. This research establishes the major principles and main elements of social sustainability, its theoretical and practical underpinnings, investigates its implementation and provides a framework for its implementation in the oil and gas industry in the Niger Delta region. Qualitative case study was employed for the study and thematic analysis was conducted on multiple sources of data. Using the oil and gas industry in the Niger Delta for empirical analysis, the research explored social sustainability through a synthesis of its major principles and elements alongside the socioeconomic characteristics of the region. The research established that social sustainability is a multidimensional concept and its major principle is the wellbeing of society within and across generations. Due to its multidimensional nature, the elements of social sustainability need to be derived on a casebycase basis for effective implementation of the concept. In the oil and gas industry in the Niger Delta, equity and social justice, partnership, employment and human capacity development, and social services and infrastructure emerged as the major elements of social sustainability. Stakeholder theory was used to frame the research and consistent with stakeholder theory, the adoption of partnership with the host communities suggests a partnership with resources to gain legitimacy. Novel approaches to social sustainability implementation such as the GMoU pioneered by Chevron, and new models of partnership based on the concept of value creation in stakeholder theory, are recommended. The research contributes to an emerging body of knowledge by showing the application and manifestations of a global concept at the local level. The findings have practical and theoretical value for practitioners and researchers of social sustainability.

A new perspective on the numerical and analytical treatment of a certain singular Volterra integral equationIn this thesis, the focus of our attention is on a certain linear Volterra integral equation with singular kernel. The equation is of great interest due to the fact that, under certain conditions, it possesses an in finite family of solutions, out of which only one has C1continuity. Numerous previous studies have been conducted and a variety of solution methods proposed. However, the emphasis has invariably been on determining just the differentiable solution. Thus, a significant gap in the research relating to this equation was identified and, therefore, our main objective here was to develop an effective solution method that allows us to approximate any chosen solution out of the infinite solution set. To this end, we converted the original integral equation into a singular differential form. Then, by applying a combination of analytical results from functional and real analysis, measure theory and the theory of Lebesgue integration, we reduced the problem to that of solving a regular initial value problem. Numerical methods were then applied and our experimental results proved that our method was highly effective, producing very accurate approximations to the true solution in a comparative study. Therefore, we feel our work here makes a significant contribution in this field of study, both from a theoretical viewpoint, as during the course of our research we established a direct relationship between the nonsmooth solutions of the integral equation and the weak solutions of our differential scheme, and in practice. Integral equations of this form arise in the study of heat conduction, diffusion and in thermodynamics. Therefore, another of our aims was to construct a method that could readily be applied in 'real world' modelling. Thus, as traditional models most often present as differential equations and, furthermore, as our method significantly simplifies the process of computing the solutions, we believe we have achieved this objective. Hence, in the final chapter, we highlight some of the ways in which our method could be adopted in order to help solve some of today's most challenging problems.

Insights into the Analysis of Fractional Delay Differential EquationsThis thesis is concerned with determining the analytic solution, using the method of steps, of the following fractional delay differential equation initial interval problem (FDDE IIP), c Dαy(s) = −y(t − τ ) for t > 0, τ > 0, 0 < α < 1, and y ∈ A1(0, T ] 0 t y(t) = ϕ(t) for t ∈ (−τ, 0] The properties of the analytic solution obtained are a surprise but they do sit comfortably when compared with those of the analytic solutions of an ordinary differential equation initial value problem (ODE IVP), a delay differential equation initial interval problem (DDE IIP) and an fractional ordinary differential equation initial value problem (FODE IVP). Further the analytic solution formula obtained is closely related to that of the analytic solution formula of the DDE IIP. However, these insights into the analytic solution of the FDDE IIP we have not seen before, and differ from those published elsewhere.

Group Codes, Composite Group Codes and Constructions of SelfDual CodesThe main research presented in this thesis is around constructing binary selfdual codes using group rings together with some wellknown code construction methods and the study of group codes and composite group codes over different alphabets. Both these families of codes are generated by the elements that come from group rings. A search for binary selfdual codes with new weight enumerators is an ongoing research area in algebraic coding theory. For this reason, we present a generator matrix in which we employ the idea of a bisymmetric matrix with its entries being the block matrices that come from group rings and give the necessary conditions for this generator matrix to produce a selfdual code over a fi nite commutative Frobenius ring. Together with our generator matrix and some wellknown code construction methods, we find many binary selfdual codes with parameters [68, 34, 12] that have weight enumerators that were not known in the literature before. There is an extensive literature on the study of different families of codes over different alphabets and speci fically finite fi elds and finite commutative rings. The study of codes over rings opens up a new direction for constructing new binary selfdual codes with a rich automorphism group via the algebraic structure of the rings through the Gray maps associated with them. In this thesis, we introduce a new family of rings, study its algebraic structure and show that each member of this family is a commutative Frobenius ring. Moreover, we study group codes over this new family of rings and show that one can obtain codes with a rich automorphism group via the associated Gray map. We extend a well established isomorphism between group rings and the subring of the n x n matrices and show its applications to algebraic coding theory. Our extension enables one to construct many complex n x n matrices over the ring R that are fully de ned by the elements appearing in the first row. This property allows one to build generator matrices with these complex matrices so that the search field is practical in terms of the computational times. We show how these complex matrices are constructed using group rings, study their properties and present many interesting examples of complex matrices over the ring R. Using our extended isomorphism, we de ne a new family of codes which we call the composite group codes or for simplicity, composite Gcodes. We show that these new codes are ideals in the group ring RG and prove that the dual of a composite Gcode is also a composite Gcode. Moreover, we study generator matrices of the form [In  Ω(v)]; where In is the n x n identity matrix and Ω(v) is the composite matrix that comes from the extended isomorphism mentioned earlier. In particular, we show when such generator matrices produce selfdual codes over finite commutative Frobenius rings. Additionally, together with some generator matrices of the type [In  Ω(v)] and the wellknown extension and neighbour methods, we fi nd many new binary selfdual codes with parameters [68, 34, 12]. Lastly in this work, we study composite Gcodes over formal power series rings and finite chain rings. We extend many known results on projections and lifts of codes over these alphabets. We also extend some known results on γadic codes over the infi nite ring R∞

Group rings: Units and their applications in selfdual codesThe initial research presented in this thesis is the structure of the unit group of the group ring Cn x D6 over a field of characteristic 3 in terms of cyclic groups, specifically U(F3t(Cn x D6)). There are numerous applications of group rings, such as topology, geometry and algebraic Ktheory, but more recently in coding theory. Following the initial work on establishing the unit group of a group ring, we take a closer look at the use of group rings in algebraic coding theory in order to construct selfdual and extremal selfdual codes. Using a well established isomorphism between a group ring and a ring of matrices, we construct certain selfdual and formally selfdual codes over a finite commutative Frobenius ring. There is an interesting relationships between the Automorphism group of the code produced and the underlying group in the group ring. Building on the theory, we describe all possible group algebras that can be used to construct the wellknown binary extended Golay code. The double circulant construction is a wellknown technique for constructing selfdual codes; combining this with the established isomorphism previously mentioned, we demonstrate a new technique for constructing selfdual codes. New theory states that under certain conditions, these selfdual codes correspond to unitary units in group rings. Currently, using methods discussed, we construct 10 new extremal selfdual codes of length 68. In the search for new extremal selfdual codes, we establish a new technique which considers a double bordered construction. There are certain conditions where this new technique will produce selfdual codes, which are given in the theoretical results. Applying this new construction, we construct numerous new codes to verify the theoretical results; 1 new extremal selfdual code of length 64, 18 new codes of length 68 and 12 new extremal selfdual codes of length 80. Using the well established isomorphism and the common four block construction, we consider a new technique in order to construct selfdual codes of length 68. There are certain conditions, stated in the theoretical results, which allow this construction to yield selfdual codes, and some interesting links between the group ring elements and the construction. From this technique, we construct 32 new extremal selfdual codes of length 68. Lastly, we consider a unique construction as a combination of block circulant matrices and quadratic circulant matrices. Here, we provide theory surrounding this construction and conditions for full effectiveness of the method. Finally, we present the 52 new selfdual codes that result from this method; 1 new selfdual code of length 66 and 51 new selfdual codes of length 68. Note that different weight enumerators are dependant on different values of β. In addition, for codes of length 68, the weight enumerator is also defined in terms of γ, and for codes of length 80, the weight enumerator is also de ned in terms of α.

Numerical methods for deterministic and stochastic fractional partial differential equationsIn this thesis we will explore the numerical methods for solving deterministic and stochastic space and time fractional partial differential equations. Firstly we consider Fourier spectral methods for solving some linear stochastic space fractional partial differential equations perturbed by spacetime white noises in one dimensional case. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. We approximate the spacetime white noise by using piecewise constant functions and obtain the approximated stochastic space fractional partial differential equations. The approximated stochastic space fractional partial differential equations are then solved by using Fourier spectral methods. Secondly we consider Fourier spectral methods for solving stochastic space fractional partial differential equation driven by special additive noises in one dimensional case. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. The spacetime noise is approximated by the piecewise constant functions in the time direction and by appropriate approximations in the space direction. The approximated stochastic space fractional partial differential equation is then solved by using Fourier spectral methods. Thirdly, we will consider the discontinuous Galerkin time stepping methods for solving the linear space fractional partial differential equations. The space fractional derivatives are defined by using Riesz fractional derivative. The space variable is discretized by means of a Galerkin finite element method and the time variable is discretized by the discontinous Galerkin method. The approximate solution will be sought as a piecewise polynomial function in t of degree at most q−1, q ≥ 1, which is not necessarily continuous at the nodes of the defining partition. The error estimates in the fully discrete case are obtained and the numerical examples are given. Finally, we consider error estimates for the modified L1 scheme for solving time fractional partial differential equation. Jin et al. (2016, An analysis of the L1 scheme for the subdiffifusion equation with nonsmooth data, IMA J. of Number. Anal., 36, 197221) ii established the O(k) convergence rate for the L1 scheme for both smooth and nonsmooth initial data. We introduce a modified L1 scheme and prove that the convergence rate is O(k2−α=), 0 < α < 1 for both smooth and nonsmooth initial data. We first write the time fractional partial differential equations as a Volterra integral equation which is then approximated by using the convolution quadrature with some special generating functions. A Laplace transform method is used to prove the error estimates for the homogeneous time fractional partial differential equation for both smooth and nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Mathematical Modelling of DNA MethylationDNA methylation is a key epigenetic process which has been intimately associated with gene regulation. In recent years growing evidence has associated DNA methylation status with a variety of diseases including cancer, Alzheimer’s disease and cardiovascular disease. Moreover, changes to DNA methylation have also recently been implicated in the ageing process. The factors which underpin DNA methylation are complex, and remain to be fully elucidated. Over the years mathematical modelling has helped to shed light on the dynamics of this important molecular system. Although the existing models have contributed significantly to our overall understanding of DNA methylation, they fall short of fully capturing the dynamics of this process. In this work DNA methylation models are developed and improved and their suitability is demonstrated through mathematical analysis and computational simulation. In particular, a linear and nonlinear deterministic model are developed which capture more fully the dynamics of the key intracellular events which characterise DNA methylation. Furthermore, uncertainty is introduced into the model to describe the presence of intrinsic and extrinsic cell noise. This way a stochastic model is constructed and presented which accounts for the stochastic nature in cell dynamics. One of the key predictions of the model is that DNA methylation dynamics do not alter when the quantity of DNA methylation enzymes change. In addition, the nonlinear model predicts DNA methylation promoter bistability, which is commonly observed experimentally. Moreover, a new way of modelling DNA methylation uncertainty is introduced.

Numerical Solution of Fractional Differential Equations and their Application to Physics and EngineeringThis dissertation presents new numerical methods for the solution of fractional differential equations of single and distributed order that find application in the different fields of physics and engineering. We start by presenting the relationship between fractional derivatives and processes like anomalous diffusion, and, we then develop new numerical methods for the solution of the timefractional diffusion equations. The first numerical method is developed for the solution of the fractional diffusion equations with Neumann boundary conditions and the diffusivity parameter depending on the space variable. The method is based on finite differences, and, we prove its convergence (convergence order of O(Δx² + Δt²<sup>α</sup>), 0 < α < 1) and stability. We also present a brief description of the application of such boundary conditions and fractional model to real world problems (heat flux in human skin). A discussion on the common substitution of the classical derivative by a fractional derivative is also performed, using as an example the temperature equation. Numerical methods for the solution of fractional differential equations are more difficult to develop when compared to the classical integerorder case, and, this is due to potential singularities of the solution and to the nonlocal properties of the fractional differential operators that lead to numerical methods that are computationally demanding. We then study a more complex type of equations: distributed order fractional differential equations where we intend to overcome the second problem on the numerical approximation of fractional differential equations mentioned above. These equations allow the modelling of more complex anomalous diffusion processes, and can be viewed as a continuous sum of weighted fractional derivatives. Since the numerical solution of distributed order fractional differential equations based on finite differences is very time consuming, we develop a new numerical method for the solution of the distributed order fractional differential equations based on Chebyshev polynomials and present for the first time a detailed study on the convergence of the method. The third numerical method proposed in this thesis aims to overcome both problems on the numerical approximation of fractional differential equations. We start by solving the problem of potential singularities in the solution by presenting a method based on a nonpolynomial approximation of the solution. We use the method of lines for the numerical approximation of the fractional diffusion equation, by proceeding in two separate steps: first, spatial derivatives are approximated using finite differences; second, the resulting system of semidiscrete ordinary differential equations in the initial value variable is integrated in time with a nonpolynomial collocation method. This numerical method is further improved by considering graded meshes and an hybrid approximation of the solution by considering a nonpolynomial approximation in the first subinterval which contains the origin in time (the point where the solution may be singular) and a polynomial approximation in the remaining intervals. This way we obtain a method that allows a faster numerical solution of fractional differential equations (than the method obtained with nonpolynomial approximation) and also takes into account the potential singularity of the solution. The thesis ends with the main conclusions and a discussion on the main topics presented along the text, together with a proposal of future work.

Insights from the parallel implementation of efficient algorithms for the fractional calculusThis thesis concerns the development of parallel algorithms to solve fractional differential equations using a numerical approach. The methodology adopted is to adapt existing numerical schemes and to develop prototype parallel programs using the MatLab Parallel Computing Toolbox (MPCT). The approach is to build on existing insights from parallel implementation of ordinary differential equations methods and to test a range of potential candidates for parallel implementation in the fractional case. As a consequence of the work, new insights on the use of MPCT for prototyping are presented, alongside conclusions and algorithms for the effective implementation of parallel methods for the fractional calculus. The principal parallel approaches considered in the work include:  A RungeKutta Method for Ordinary Differential Equations including the application of an adapted Richardson Extrapolation Scheme  An implementation of the DiethelmChern Algorithm for Fractional Differential Equations  A parallel version of the wellestablished Fractional Adams Method for Fractional Differential Equations  The adaptation for parallel implementation of Lubich's Fractional Multistep Method for Fractional Differential Equations An important aspect of the work is an improved understanding of the comparative diffi culty of using MPCT for obtaining fair comparisons of parallel implementation. We present details of experimental results which are not satisfactory, and we explain how the problems may be overcome to give meaningful experimental results. Therefore, an important aspect of the conclusions of this work is the advice for other users of MPCT who may be planning to use the package as a prototyping tool for parallel algorithm development: by understanding how implicit multithreading operates, controls can be put in place to allow likeforlike performance comparisons between sequential and parallel programs.

Higher Order Numerical Methods for Fractional Order Differential EquationsThis thesis explores higher order numerical methods for solving fractional differential equations.

Numerical treatment of oscillatory delay and mixed functional differential equations arising in modellingThe pervading theme of this thesis is the development of insights that contribute to the understanding of whether certain classes of functional differential equation have solutions that are all oscillatory. The starting point for the work is the analysis of simple (linear autonomous) ordinary differential equations where existing results allow a full explanation of the phenomena. The Laplace transform features as a key tool in developing a theoretical background. The thesis goes on to explore the corresponding theory for delay equations, advanced equations and functional di erential equations of mixed type. The focus is on understanding the links between the characteristic roots of the underlying equation, and the presence or otherwise of oscillatory solutions. The linear methods are used as a class of numerical schemes which lead to discrete problems analogous to each of the classes of functional differential equation under consideration. The thesis goes on to discuss the insights that can be obtained for discrete problems in their own right, and then considers those new insights that can be obtained about the underlying continuous problem from analysis of the oscillatory behaviour of the analogous discrete problem. The main conclusions of the work are some semiautomated computational approaches (based upon the Principle of the Argument) which allow the prediction of oscillatory solutions to be made. Examples of the effectiveness of the approach are provided, and there is some discussion of its theoretical basis. The thesis concludes with some observations about further work and some of the limitations of existing analytical insights which restrict the reliability with which the approach developed can be applied to wider classes of problem.

Computational and mathematical modelling of plant species interactions in a harsh climateThis thesis will consider the following assumptions which are based on a few insights about the artic climate: (1)the artic climate can be characterised by a growing season called summer and a dormat season called winter (2)in the summer season growing conditions are reasonably favourable and species are more likely to compete for plentiful resources (3)in the winter season there would be no further growth and the plant populations would instead by subjected to fierce weather events such as storms which is more likely to lead to the destruction of some or all of the biomass. Under these assumptions, is it possible to find those change in the environment that might cause mutualism (see section 1.9.2) from competition (see section 1.9.1) to change? The primary aim of this thesis to to provide a prototype simulation of growth of two plant species in the artic that: (1)take account of different models for summer and winter seasons (2)permits the effects of changing climate to be seen on each type of plant species interaction.

The numerical solution of fractional and distributed order differential equationsFractional Calculus can be thought of as a generalisation of conventional calculus in the sense that it extends the concept of a derivative (integral) to include noninteger orders. Effective mathematical modelling using Fractional Differential Equations (FDEs) requires the development of reliable flexible numerical methods. The thesis begins by reviewing a selection of numerical methods for the solution of Singleterm and Multiterm FDEs. We then present: 1. a graphical technique for comparing the efficiency of numerical methods. We use this to compare Singleterm and Multiterm methods and give recommendations for which method is best for any given FDE. 2. a new method for the solution of a nonlinear Multiterm Fractional Dif¬ferential Equation. 3. a sequence of methods for the numerical solution of a Distributed Order Differential Equation. 4. a discussion of the problems associated with producing a computer program for obtaining the optimum numerical method for any given FDE.

Noise induced changes to dynamic behaviour of stochastic delay differential equationsThis thesis is concerned with changes in the behaviour of solutions to parameterdependent stochastic delay differential equations.

Numerical analysis of some integral equations with singularitiesIn this thesis we consider new approaches to the numerical solution of a class of Volterra integral equations, which contain a kernel with singularity of nonstandard type. The kernel is singular in both arguments at the origin, resulting in multiple solutions, one of which is differentiable at the origin. We consider numerical methods to approximate any of the (infinitely many) solutions of the equation. We go on to show that the use of product integration over a short primary interval, combined with the careful use of extrapolation to improve the order, may be linked to any suitable standard method away from the origin. The resulting splitinterval algorithm is shown to be reliable and flexible, capable of achieving good accuracy, with convergence to the one particular smooth solution.

Delay differential equations: Detection of small solutionsThis thesis concerns the development of a method for the detection of small solutions to delay differential equations. The detection of small solutions is important because their presence has significant influence on the analytical prop¬erties of an equation. However, to date, analytical methods are of only limited practical use. Therefore this thesis focuses on the development of a reliable new method, based on finite order approximations of the underlying infinite dimen¬sional problem, which can detect small solutions. Decisions (concerning the existence, or otherwise, of small solutions) based on our visualisation technique require an understanding of the underlying methodol¬ogy behind our approach. Removing this need would be attractive. The method we have developed can be automated, and at the end of the thesis we present a prototype Matlab code for the automatic detection of small solutions to delay differential equations.