Theses
This collection contains the Doctoral and Masters by Research theses produced within the department.
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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.