Applying the method of integration of ordinary differential equations based on the Chebyshev series to the restricted plane circular three-body problem

An approximate method of solving the Cauchy problem for nonlinear first-order ordinary differential equations is considered. The method is based on using the shifted Chebyshev series and a Markov quadrature formula. An algorithm is briefly discussed to partition the integration interval into elementary subintervals where an approximate solution is represented by partial sums of shifted Chebyshev series with a prescribed accuracy. The efficiency of the proposed method is illustrated by solving the following problem of celestial mechanics: the plane circular restricted three-body problem. The reliability of the used error estimate and its proximity to the true error are shown. A number of advantages of the proposed method over the well-known Gear method of solving ordinary differential equations are analyzed.