Introduction

By a numerical solution of an ordinary differential equation

$y' = f(t,y),\hspace{1.5cm} y(t_0) = y_0\hfill(1)$

we understand the finding of a sequence:

$[y(t_0)=y_0],\hspace{0.5cm} [y(t_1)=y_1],\hspace{0.5cm} [y(t_2)=y_2], ..., [y(t_n)=y_n].$

The best-known and most accurate method of calculating a new value of a numerical solution of a differential equation (1) is to construct the Taylor series in the form

$y_{n+1} = y_n + h*f(t_n,y_n) +\frac{ h^{2}}{2!}*f^{[1]}(t_n,y_n) +\ldots +\frac{ h^{p}}{p!}*f^{[p-1]}(t_n,y_n),\hfill (2)$

where h is the integration step.

Methods of numerical solutions of differential equations have been studied since the end of the last century. A large number of integration formulas have been published especially for solving special systems of differential equations. In general, it was not possible to choose the best method but for a subclass of tasks defined by similar properties the most suitable method could always be found.

The presented " Modern Taylor Series Method " has proved to be both very accurate and fast. It is based on a direct use of the Taylor series.

The main idea behind the Modern Taylor Series Method is an automatic integration method order setting, i.e. using as many Taylor series terms for computing as needed to achieve the required accuracy.