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.

The synthesis and biological evaluation of lanostane and cholestane-type natural products

The main objective of this thesis is to outline the synthetic chemistry involved in the preparation of a range of novel lanostane and cholestane derivatives, and subsequent investigation into their biological activity in cancer cells. The biological results obtained throughout the project have driven the strategic synthesis of new compounds, in an effort to optimise the anti cancer potential of lanostane and cholestane derivatives. The first chapter begins with an overview of steroidal compounds and details a literature review of the natural sources of these moieties, as well as their biosynthesis and reported synthetic derivatives. The biological activity of interesting natural and synthetic analogues is also discussed. In addition, an insight into some currently prescribed pharmaceutical compounds, with functional groups relevant to this project, is presented. The second chapter discusses the methods employed for the synthesis of these novel lanostane and cholestane derivatives, and comprises three main sections. Firstly, various oxidation products of lanosterol are synthesised, mainly via epoxidations of the C-8,9 and C- 24,25 alkenes, and also allylic oxidations at these positions. Secondly, amine derivatives of lanosterol are formed by cleaving the lanostane side chain, thereby yielding a new cholestane nucleus, and performing several reductive aminations on the resulting key aldehyde intermediates. Various amines such as piperidine, morpholine, diethylamine and aniline are employed in the reductive amination reactions to yield novel cholestane steroids with amine side chains. Finally, starting from stigmasterol and proceeding with the same methodology of cleaving the steroidal side chain and subsequently performing reductive aminations, novel cholestane derivatives of the biologically active amines are synthesised. The cytotoxicity of these compounds against CaCo-2 and U937 cell lines is presented in terms of percentage viability of cells, IC50 value and apoptosis. The MTT assay is used to determine the percentage viability of cells, and the IC50 data is generated from the MTT results. Apoptosis is measured in terms of fold increase relative to a carrier control. In summary, the compounds formed are discussed in terms of chemical synthesis, spectroscopic interpretation and biological activity. The main reaction pathways involved in the chemistry within this project are various oxidations and reductive amination. The final chapter is a detailed account of the full experimental procedures for the compounds synthesised during this work, including characterisation using spectroscopic and analytical data.