Approximate solution methods for the Cauchy problem with the use of solution operators and the taylor formula

Abstract

To find an approximate solution of the Cauchy problem

$$\dot x(t) = f(x(t),t),\;\;\;\;\;x(t_0 ) = x^0 \in R^n ,\;t \geqslant t_0 ,$$

we propose numerical methods of the fourth and fifth order of accuracy that are based on solution operators and the expansion of the solution of the Cauchy problem into a Taylor series. Results of numerical experiments are given. Bibliography: 3 titles.