| Summary: | Unconventional methods for the numerical solution of first order initial value problems (IVPs) are well established in the past decades. There are two major reasons that motivate the developments of unconventional methods: firstly, unconventional methods are developed to solve certain types of IVPs, such as IVPs with oscillatory solutions or IVPs whose solutions possess singularities, where in most of the time, conventional methods will perform poorly; and secondly, unconventional methods might possess some outstanding features that could never be achieved by conventional methods. These features include achieving high order of numerical accuracy with less computational cost, stronger stability properties and so on. In this thesis, some studies are made of the unconventional methods based on rational functions and mean expressions. The study has led to the discovery of some new exponential-rational methods and rational multistep methods which can be used effectively for numerical solution of first order IVPs. The study continues with the discoveries of some new pseudo Runge-Kutta methods based on harmonic and arithmetic means; and a multistep method based on centroidal mean, which are found to be effective for the numerical solution of first order IVPs. This thesis also includes the study of implicit Runge-Kutta (IRK) methods, which led to the developments of three new classes of IRK methods based on Kronrod-type quadrature formulae. Each new method developed in this thesis is furnished with local truncation error and absolute stability analysis. In addition, each new method is tested on some test problems and also compared with other conventional or classical methods in the same order.
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