This work is designed to transform the fifth 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 is because the rooted trees and derived equations for the y derivatives and x,y derivatives are the same for the explicit fourth-stage fourth-order methods, hence, we are motivated to analyze the fifth-stage fourth-order method. This idea is also derivable from general graphs and combinatorics.
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