Application of Chebyshev series to integration of ordinary differential equations with rapidly growing solutions

A method of solving systems of ordinary differential equations is described. This method is based on the approximation of right-hand sides by partial sums of shifted Chebyshev series. The coefficients of the series are determined using Markov quadrature formulas. It is shown that the proposed method is more efficient compared to the Runge–Kutta and Adams methods when solving differential equations with rapidly growing solutions.