On modifications of Runge–Kutta–Nyström methods for solving y(4)=f(x,y)y(4)=f(x,y)

We present Runge–Kutta–Nyström type methods for the solution of a special fourth order initial value problem. Order conditions theory is given based on rooted trees. Thus a new set of SN-trees is introduced named ⊤4 and the enumeration of its elements is studied. A Mathematica package is also listed that may furnish instantly order conditions of very high orders. Finally a new 8th order method is constructed using Differential Evolution for solving the equations of condition. It outperforms by far the well-known established methods.