Modification of Fourth order Runge-Kutta Method for Kutta Form With Geometric Means

This paper  discuss how to modified Fourth order Runge-Kutta Kutta method based on the geometric mean. Then we have parameters  and   however by re-comparing the Taylor series expansion of  and  up to the 4th order.  For make error term re-compering of  the Taylor series expansion of  and  up to the 5th order. In the error term an make substitution for the values of  and  into the Taylor seriese expansion up to the 5th order. So that we have error term modified Fourth Order Runge-Kutta Kutta based on the geometric mean.  Modified Fourth Order Runge-Kutta Kutta based on the geometric mean that usually used to solved ordinary differential equations.