Theseshttp://hdl.handle.net/10034/6230532020-08-04T06:57:30Z2020-08-04T06:57:30ZInsights from the parallel implementation of efficient algorithms for the fractional calculusBanks, Nicola E.http://hdl.handle.net/10034/6138412020-04-02T09:11:28Z2015-07-01T00:00:00ZInsights from the parallel implementation of efficient algorithms for the fractional calculus
Banks, Nicola E.
This 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 Runge-Kutta Method for Ordinary Differential Equations including the application of an adapted Richardson Extrapolation Scheme - An implementation of the Diethelm-Chern Algorithm for Fractional Differential Equations - A parallel version of the well-established 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 like-for-like performance comparisons between sequential and parallel programs.
2015-07-01T00:00:00ZHigher Order Numerical Methods for Fractional Order Differential EquationsPal, Kamalhttp://hdl.handle.net/10034/6133542020-04-02T09:11:45Z2015-08-01T00:00:00ZHigher Order Numerical Methods for Fractional Order Differential Equations
Pal, Kamal
This thesis explores higher order numerical methods for solving fractional differential equations.
2015-08-01T00:00:00ZNumerical treatment of oscillatory delay and mixed functional differential equations arising in modellingMalique, Md A.http://hdl.handle.net/10034/3110002020-04-02T09:12:06Z2012-09-01T00:00:00ZNumerical treatment of oscillatory delay and mixed functional differential equations arising in modelling
Malique, Md A.
The 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 semi-automated 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.
2012-09-01T00:00:00ZComputational and mathematical modelling of plant species interactions in a harsh climateEkaka-A, Enu-Obari N.http://hdl.handle.net/10034/1180162020-04-02T09:13:31Z2009-07-01T00:00:00ZComputational and mathematical modelling of plant species interactions in a harsh climate
Ekaka-A, Enu-Obari N.
This 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.
2009-07-01T00:00:00Z