Fluctuations are known to radically alter the behaviour of reaction-diffusion systems. Below a certain upper critical dimension d_c , this effect results in the breakdown of traditional approaches, such as mean field rate equations. In this thesis we tackle this fluctuation problem by employing systematic field theoretic/renormalisation group methods, which enable perturbative calculations to be made below d_c. We first consider a steady state reaction front formed in the two species irreversible reaction A + B → Ø. In one dimension we demonstrate that there are two components to the front - one an intrinsic width, and one caused by the ability of the centre of the front to wander. We make theoretical predictions for the shapes of these components, which are found to be in good agreement with our one dimensional simulations. In higher dimensions, where the intrinsic component dominates, we also make calculations for its asymptotic profile. Furthermore, fluctuation effects lead to a prediction of asymptotic power law tails in the intrinsic front in all dimensions. This effect causes high enough order spatial moments of a time dependent reaction front to exhibit multiscaling. The second system we consider is a time dependent multispecies reaction-diffusion system with three competing reactions A+A → Ø, B + B → Ø, and A + B → Ø, starting with homogeneous initial conditions. Using our field theoretic formalism we calculate the asymptotic density decay rates for the two species for d ≤ d_c. These calculations are compared with other approximate methods, such as the Smoluchowski approach, and also with previous simulations and exact results.

The primary purpose of the work undertaken in this thesis is to investigate soliton scattering in the non linear CP² sigma model. This has two spatial and one temporal dimension. The vector fields used to represent the model have three components and hence there exists a global SU(3) symmetry. The effects of adding an Hopflike term to the basic lagrangian is considered. A review of the model is given in chapter I. The second chapter discusses Noether's theorem which states that each symmetry of the lagrangian has associated with it a conserved charge. In the third chapter, the eight charges relating to the internal symmetry are calculated. Explanations are provided for the results calculated during the numerical simulations. The results for the CP¹ model are also discussed. In the fourth chapter, these charges are used to predict the qualitative behaviour of the solitons. It will provide an explanation for the effect of the coefficient of the hopflike term on the scattering. The single soliton ansatz is also investigated. In the penultimate chapter, an alternative approach is used. This involves looking for the closest static approximation to the evolved solution. It is able to predict the trajectory for pure CP² and some confirmation is provided for the ansatz used in the full lagrangian. The last chapter summarises the results. It also provides some suggestions for further work.

The author studies two coupled reaction-diffusion equations of 'λ-w' type, on an open, bounded, convex domain Ω C R(^d) (d ≤ 3), with a boundary of class C², and homogeneous Neumann boundary conditions. The equations are close to a supercritical Hopf bifurcation in the reaction kinetics, and are model equations for oscillatory reaction-diffusion equations. Global existence, uniqueness and continuous dependence on initial data of strong and weak solutions are proved using the classical Faedo-Galerkin method of Lions and compactness arguments. The work provides a complete case study for the application of this method to systems of nonlinear reaction-diffusion equations. The author also undertook the numerical analysis of the reaction-diffusion system. Results are presented for a fully-practical piecewise linear finite element method by mimicking results in the continuous case. Semi-discrete and fully-discrete error estimates are proved after establishing a priori bounds for various norms of the approximate solutions. Finally, the theoretical results are illustrated and verified via the numerical simulation of periodic plane waves in one space dimension, and preliminary results representing target patterns and spiral solutions presented in two space dimensions.

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This 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.

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This thesis presents methodology to analyse repeated ordered categorical data (repeated ordinal data), under the assumption that measurements arise as discrete realisations of an underlying (latent) continuous distribution. Two sets of estimation equations, called quasiestimation equations or QEEs, are presented to estimate the mean structure and the cutoff points which define boundaries between different categories. A series of simulation studies are employed to examine the quality of the estimation processes and of the estimation of the underlying latent correlation structure. Graphical studies and theoretical considerations are also utilised to explore the asymptotic properties of the correlation, mean and cutoff parameter estimates. One important aspect of repeated analysis is the structure of the correlation and simulation studies are used to look at the effect of correlation misspecification, both on the consistency of estimates and their asymptotical stability. To compare the QEEs with current methodology, simulations studies are used to analyse the simple case where the data are binary, so that generalised estimation equations (GEEs) can also be applied to model the latent trend. Again the effect of correlation misspecification will be considered. QEEs are applied to a data set consisting of the pain runners feel in their legs after a long race. Both ordinal and continuous responses are measured and comparisons between QEEs and continuous counterparts are made. Finally, this methodology is extended to the case when there are multivariate repeated ordinal measurements, giving rise to inter-time and intra-time correlations.

Recent surveys have revealed that the majority of numerical methods for the solution of integral equations use one of two main techniques for generating a set of simultaneous equations for their solution. Either the unknown function is expanded as a combination of basis set functions and the resulting coefficients found, or the integral is discretized using quadrature formulae. The latter results in simultaneous equations for the solution at the quadrature abscissae. The thesis proposes techniques based on various direct iterative methods, including refinements of residual correction which hold no restrictions for nonlinear integral equations. New implementations of successive approximations and Newton's method appear. The latter compares particularly well with other versions as the evaluation of the Jacobian can be made equivalent to the solution of matrix equations of relatively small dimensions. The method can be adapted to the solution of first-kind equations and has been applied to systems of integral equations. The schemes are designed to be adaptive with the aid of the progressive quadrature rules of Patterson or Clenshaw and Curtis and interpolation formulae. The Clenshaw-Curtis rule is particularly favoured as it delivers error estimates. A very powerful routine for the solution of a wide range of integral equations has resulted with the inclusion of a new efficient method for calculating singular integrals. Some work is devoted to the conversion of differential to integral or integro-differential equations and comparing the merits of solving a problem in its original and converted forms. Many equations are solved as test examples throughout the thesis of which several are of physical significance. They include integral equations for the slowing down of neutrons, the Lane-Emden equation, an equation arising from a chemical reactor problem, Chandrasekhar's isotropic scatter ing of radiation equation and the Blasius equation in boundary layer theory.

This thesis is concerned with an investigation and evaluation of collocation algorithms for solving two-point boundary value problems for systems of ordinary differential equations. An emphasis is on developing reliable and efficient adaptive mesh selection algorithms in piecewise collocation methods. General background materials including basic concepts and descriptions of the method as well as some functional analysis tools needed in developing some error estimates are given at the beginning. A brief review of some developments in the methods to be used is provided for later referencing. By utilising the special structure of the collocation matrices, a more compact block matrix structure is introduced and an algorithm for generating and solving the matrix is proposed. Some practical aspects and computational considerations of matrices involved in the collocation process such as analysis of arithmetic operations and amount of memory spaces needed are considered. An examination of scaling process to reduce the condition number is also presented. A numerical evaluation of some error estimates developed by considering the differential operator, the related matrices and the residual is carried out. These estimates are used to develop adaptive mesh selection algorithms, in particular as a cheap criterion for terminating the computation process. Following a discussion on mesh selection strategies, a criterion function for use in adaptive algorithms is introduced and a numerical scheme to equidistributing values of the criterion function is proposed. An adaptive algorithm based on this criterion is developed and the results of numerical experiments are compared with those using some well known criterion functions. The various examples are chosen in such a way that they include problems with interior or boundary layers. In addition, an algorithm has been developed to predict the necessary number of subintervals for a given tolerance, with the aim of improving the efficiency of the whole process. Using a good initial mesh in adaptive algorithms would be expected to provide some further improvement in the algorithms. This leads to the idea of locating the layer regions and determining suitable break points in such regions before the numerical process. Based on examining the eigenvalues of the coefficient matrix in the differential equation in the specified interval, using their magnitudes and rates of change, the algorithms for predicting possible layer regions and estimating the number of break points needed in such regions are constructed. The effectiveness of these algorithms is evaluated by carrying out a number of numerical experiments. The final chapter gives some concluding remarks of the work and comment on results of numerical experiments. Certain possible improvements and extensions for further research are also briefly given.