An implementation of the Chebyshev series method for the approximate analytical solution of second-order ordinary differential equations

A method used to apply the Chebyshev series for solving canonical systems of second order ordinary differential equations is described. This method is based on the approximation of the Cauchy problem solution and its first and second derivatives by partial sums of shifted Chebyshev series. The coefficients of these series are determined iteratively using the relations relating the Chebyshev coefficients of the solution and its first derivative with the Chebyshev coefficients found for the right-hand side of the canonical system by application of Markovs quadrature formula. The obtained numerical results are discussed and the high-precision analytical representations of the solution are proposed in the form of partial sums of Chebyshev series on a given integration segment.