New numerical strategies for initial value type ordinary differential equations
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.