Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Chemistry of B1- and B10- boranes containing halogen or pseudohalogen groups

The research described in this thesis involved the chemistry of borane-species which contain one or more halide or pseudohalide groups. Both monoboron species e.g. [BH3X]- and "cluster" borane species e.g. [B10H9X]2- and I-Se B11H10 were studied. The first chapter is a review of the syntheses, properties and reactions of halide and pseudohalide species containing from one to ten boron atoms. Chapter Two is a theoretical investigation of' the electronic and molecular structures of two series of boranes i. e. [BH3X]- and [B10H9X]2- where X = H, CI, CN, NCS, SCN and N3. The calculational method used was the Modified Neglect of Differential Overlap (MNDO) method of Dewar et al. The results were compared where possible with experimental results such as the X-ray crystallographically determined structures of [BH3CI]- and [B10H10]2-. Chapter Three concerns halogenated selenaborane clusters and reports an improved synthesis of 12-Br-SeB11H10 and the first structural data for a simple non-metal containing selenaborane cage with the X-ray crystallographically determined structure of 12-1-SeB11H10. Finally, an indepth n.m.r. study of Se2B9H9 is also reported together with attempts to halogenate this compound. The last two chapters are based on single boron systems. Chapter Four concerns the synthetic routes to amine-boranes and -cyanoboranes from [BH4]- and [BH3CN]- substrates. This chapter discusses some difficulties encountered when polyamines were used in these reactions. The characterisation of an unusual ketone isolated from some of these reactions, the X-ray crystallographically determined structure of 4-dimethylamino-pyridine-cyanoborane and a new route to pyrazabole dimeric species are also discussed. The final chapter reports on work carried out at producing BH2X (X = H, CN) adducts of aminophosphines. Three routes were attempted to generate P-B and N-B bonded species with varying degrees of success. Some unusual products of these reactions are discussed including [Ph2(O) PPPh2 ] [Ph2NH]2, the structure of which was determined by X-ray crystallography.