On The Component Analysis and Transformation of An Explicit Fourth – Stage Fourth – Order Runge – Kutta Methods

Abstract

This work is designed to transform the fourth stage – fourth order explicit Runge-Kutta method with the aim of projecting a new method of implementing it through tree diagram analysis. Efforts will be made to represent the equations derived from the y derivatives and x,y derivatives separately on Butcher’s rooted trees. This idea is derivable from general graphs and combinatorics. Key words: Rooted tree diagram, Transformation, Vertex, explicit, y partial derivatives, x,y partial derivatives, Runge-Kutta Methods, Linear and non- linear equations, Taylor series, Graphs. Keywords: key words, Runge - Kutta, Combinatorics, Derivatives, Transformation, Analysi