Application of Chebyshev series for the integration of ordinary differential equations

A numerical analytic method is proposed for solving the Cauchy problem for normal systems of ordinary differential equations. This method is based on the approximation of the solution and its derivative by partial sums of shifted Chebyshev series. The coefficients of the series are determined by with the aid of an iterative process using Markov's quadrature formulas. The method yields an analytical representation of a solution and can be used to solve ordinary differential equations with a higher accuracy and with a larger discretization step compared to the classical methods, such as Runge–Kutta, Adams, and Gear methods.