Mathematics

Application of Chebyshev series to numerical integration of nonlinear oscillation equation

S. F. Zaletkin

Lomonosov Moscow State University, Research Computing Center

Abstract: An approximate method of solving the Cauchy problem for nonlinear second-order ordinary differential equations is considered. The method is based on conforming 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 and its derivative are represented by partial sums of shifted Chebyshev series with a prescribed accuracy. The efficiency of the proposed method is illustrated by solving nonlinear oscillations of mathematical pendulum equation. A number of advantages of the proposed method over the well-known Stoermen method of solving ordinary differential equations are analyzed.

Key words: ordinary differential equations, approximate analytical methods, numerical methods, orthogonal expansions, shifted Chebyshev series, Markov quadrature formulas, polynomial approximation, precision control, error estimate, automatic step size control.

UDC: 519.622

Received: 12.01.2024

DOI: 10.55959/MSU0579-9368-1-66-3-2

 English version: Moscow University Mathematics Bulletin, Moscow University Mеchanics Bulletin, 2025, 80:3, 160–166

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