Methods and algorithms of computational mathematics and their applications

Approximate integration of the Cauchy problem for canonical systems of second order ordinary differential equations by the Chebyshev series method with precision control

S. F. Zaletkin

Lomonosov Moscow State University, Research Computing Center

Abstract: An approximate method of solving the Cauchy problem for canonical systems of second order ordinary differential equations is considered. The method is based on using the shifted Chebyshev series and a Markov quadrature formula. Some approaches are given to estimate the errors of an approximate solution and its derivative expressed by partial sums of a certain order shifted Chebyshev series. The errors are estimated using the second approximation of the solution calculated in a special way and expressed by a partial sum of a higher order series. An algorithm of partitioning the integration interval into elementary subintervals to ensure the computation of the solution and its derivative with prescribed accuracy is discussed on the basis of proposed approaches to error estimation.

Keywords: ordinary differential equations, approximate analytic methods, numerical methods, orthogonal expansions, shifted Chebyshev series, Markov's quadrature formulas, polynomial approximation, precision control, error estimate, automatic step size control.

UDC: 519.622

Received: 06.03.2025
Accepted: 15.04.2025

DOI: 10.26089/NumMet.v26r212