ChesterRep Collection:
http://hdl.handle.net/10034/31839
Mon, 30 May 2016 10:38:41 GMT2016-05-30T10:38:41ZThe 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/10034/3456762014-09-01T00:00:00ZNumerical methods for space-fractional partial differential equations
http://hdl.handle.net/10034/345678
Title: Numerical methods for space-fractional partial differential equations
Authors: Mukta, Hasna K.
Abstract: In this dissertation, we consider numerical methods for solving space-fractional PDEs. We first consider finite difference method then we consider finite element methods for solving
space-fractional PDEs. The error estimates are obtained. Finally we consider the matrix transform technique (MTT) for solving space-fractional PDEs which include finite difference
and finite element method. Numerical examples are given.Mon, 01 Sep 2014 00:00:00 GMThttp://hdl.handle.net/10034/3456782014-09-01T00:00:00ZComparison of two numerical methods for stochastic delay differential equations and the relationship between bifurcation approximations and step length
http://hdl.handle.net/10034/345679
Title: Comparison of two numerical methods for stochastic delay differential equations and the relationship between bifurcation approximations and step length
Authors: Edmunds, Nia
Abstract: We give introductions to delay differential equations, stochastic differential equations,
numerical approximations, Brownian motion and Ito calculus, stability and bifurcation points and Lyapunov exponents. Using these methods we replicate the calculations in the paper by Neville J. Ford & Stewart J. Norton, entitled Noise induced changes to the behaviour of semi implicit Euler methods for stochastic delay differential equations undergoing bifurcation . We present our results that correspond to some of the tables and equations presented in their paper. We then apply the same methodology using a Milstein numerical method with the same parameters and random distributions and compare these results with our ndings from the Euler-Maruyama scheme. We fi nd that the Milstein scheme exhibits the same relational behaviours between the bifurcation approximations from the Lyapunov exponents and step length as was presented in Ford and Norton's paper for the Euler-Maruyama scheme, we also find that the Milstein scheme maintains its greater accuracy up to and including the bifurcation approximation.Fri, 26 Sep 2014 00:00:00 GMThttp://hdl.handle.net/10034/3456792014-09-26T00:00:00Z