Transformation and Implementation of a Highly Efficient Fully Implicit Fourth-Order Runge-Kutta Method

Abstract

The essence of this work is to transform, derive and implement a fully implicit fourth-order Runge-Kutta method of the third-stage that is highly efficient with a view of adopting a new approach. This new approach is done by separating the f(y) functional derivatives and their derived equations from the f(x,y) functional derivatives and their derived equations. Efforts will be made to vary the parameters with the aim of getting a fully implicit fourth order formula that can improve results when implemented on ordinary differential equations (ODEs). Efforts will also be made to compare both functional derivatives and their derived equations to see how related they are.