ChesterRep Collection:
http://hdl.handle.net/10034/31839
Sun, 26 Jun 2016 08:08:11 GMT2016-06-26T08:08:11ZInsights from the parallel implementation of efficient algorithms for the fractional calculus
http://hdl.handle.net/10034/613841
Title: Insights from the parallel implementation of efficient algorithms for the fractional calculus
Authors: Banks, Nicola E.
Abstract: 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.Wed, 01 Jul 2015 00:00:00 GMThttp://hdl.handle.net/10034/6138412015-07-01T00:00:00ZHigher Order Numerical Methods for Fractional Order Differential Equations
http://hdl.handle.net/10034/613354
Title: Higher Order Numerical Methods for Fractional Order Differential Equations
Authors: Pal, Kamal K.
Abstract: This thesis explores higher order numerical methods for solving fractional differential equations.Sat, 01 Aug 2015 00:00:00 GMThttp://hdl.handle.net/10034/6133542015-08-01T00:00:00ZThe elegance of differential forms in vector calculus and electromagnetics
http://hdl.handle.net/10034/345818
Title: The elegance of differential forms in vector calculus and electromagnetics
Authors: Parkinson, Christian
Abstract: In the chapter one of this text we give an introduction to, and discuss the main integral
theorems, of vector calculus; Green's theorem, Stokes' theorem and Gauss' Divergence theorem. Note that the main resource used for this chapter is [8]. Chapter two introduces
differential forms and exterior calculus; in it we discuss exterior multiplication and exterior differentiation giving proofs for properties of both. We discuss the integration of differential forms in chapter three and provide definitions of the Divergence, Gradient and Curl and main integral theorems of vector calculus including the Generalised Stokes' theorem that encloses them all in terms of such forms. Further we give a proof of the Generalised Stokes', Green's, Stokes' and Gauss' Divergence theorems. Given the elegance of differential forms that enables us to write the integral theorems of vector calculus as one theorem, the Generalised Stokes' theorem, we show a second elegance by deducing and proving Maxwell's equations, whilst reducing them from four equations to just two. Finally we provide some current research involving differential forms.Mon, 01 Sep 2014 00:00:00 GMThttp://hdl.handle.net/10034/3458182014-09-01T00:00:00ZMathematical modelling of mutualism in population ecology
http://hdl.handle.net/10034/345676
Title: Mathematical modelling of mutualism in population ecology
Authors: Rowntree, Andrew
Abstract: This research dissertation focuses on the symbiotic interaction of mutualism, we give explanations as to what it is before mathematically modelling population dynamics of two species displaying mutualistic behaviour. Throughout the course of this dissertation, we
shall be re-examining the work done in the book by Kot [16] and the paper by Joharjee and Roberts [32], whilst providing further explanations of the mathematics involved and
the steps taken. We begin by constructing a model for mutualism before attempting to improve the model in order to make it more realistic. We go on to add delays to our improved
model and determine the stability of its equilibrium points. We formulate models via piecewise constant arguments and via a simple Euler scheme before determining stability for both systems. A graphical comparison will then be made to explain the differences in behaviour between the two discretised systems.Mon, 01 Sep 2014 00:00:00 GMThttp://hdl.handle.net/10034/3456762014-09-01T00:00:00Z