Modified Runge-Kutta method

Modern Runge-Kutta method of solving ODE bears a slight resemblance with the classical (explicit) method and is based on the transformation of the differential equation to the integral one. The contents of the mathematical theory was formulated by J.C. Butcher et all. Nevertheless, the technique of constracting fundamental equations of RK-method remained unchanged. In this paper new principles are lying in the basis of constructing fundamental equations. Some new ideas are used for solving the fundamental equations, in particular, the principle of nilpotency for explicit, diagonal and singly-implicit RK-methods is successively performed.