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This thesis is concerned with the development of new numerical techniques for solving initial value problems in ordinary differential equations (ODE). The thesis begins with an introductory chapter on initial value type problems in ordinary differential equations followed by a chapter on basic mathematical concepts, which introduces and discusses, among others, the theory of Arithmetic and Geometric Means. This is followed, in Chapter 3, by a survey of the existing ODE solvers and their theoretical background. The advantages and disadvantages of some different strategies in terms of stability and truncation error are also considered. The presentation of the elementary methods based on Arithmetic Mean (AM) and Geometric Mean (GM) formulae is done in Chapter 4, with emphasis on establishing the GM trapezoida1 formula, and to the study of its stability and truncation error. Applications in the predictorcorrector and the extrapolation techniques are also considered. Special application in the solution of delay differential equations is also presented. In Chapter 5, the application of the GM strategy in the Runge-Kutta type formulae is considered, producing a new class of methods called the GM-Runge-Kutta formulae which is found to be as competitive as the classical Runge-Kutta methods. Thereafter, a new strategy of error control called the Arithmeto-Geometric Mean (AGM) strategy is developed. Further application of the GM-Runge-Kutta in Fehlberg type formulae, and the GM-Iterative Multistep formulae are also considered. Chapter 6 concerns with further applications of GM techniques in the development of generalised GM mu1tistep and multiderivative methods, and for solving y'=λ(x)y. The general idea of the GM are also extended to other types of Means, such as Harmonic and Logarithmic Means. In Chapter 7, some new formulae for solving problems with oscillatory and periodic solutions are considered. Finally the thesis concludes with recommendations for further work.

This thesis is concerned with the evaluation of rapidly oscillatory integrals, that is integrals in which the integrand has numerous local maxima and minima over the range of integration. Three numerical integration rules are presented. The first is suitable for computing rapidly oscillatory integrals with trigonometric oscillations of the form f(x) exp(irq(x)). The method is demonstrated, empirically, to be convergent and numerically stable as the order of the formula is increased. For other forms of oscillatory behaviour, a second approach based on Lagrange's identity is presented. The technique is suitable for any oscillatory weight function, provided that it satisfies an ordinary linear differential equation of order m :2:: 1. The method is shown to encompass Bessel oscillations, trigonometric oscillations and Fresnel oscillations, and products of these terms. Examples are included which illustrate the efficiency of the method in practical applications. Finally, integrals where the integrand is singular and oscillatory are considered. An extended Clenshaw-Curtis formula is developed for Fourier integrals which exhibit algebraic and logarithmic singularities. An efficient algorithm is presented for the practical implementation of the method.

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