Some Advantages of the Hybrid Methods, Which Used the First Derivative of the Solution of the Considered Problem

Abstract

The scientists began investigate of the solution of ODE from the XVII century. Many famous mathematicians as Newton, Leibniz, Bernoulli, D’ Alembert, Euler, Cauchy and etc. have considered solving of the ordinary differential equations. For solving these equations the scientists from the different country have constructed many methods. Here, with the help of the compare known results are shown some advantages of the hybrid methods. Are constructed the hybrid methods with the order of accuracy p = 8k . Here, is suggested concrete hybrid method of the fractional step type with the order of accuracy p = 8 for k =1 for which uses the first order derivative of the solution of the initial value problem.