DOI: https://doi.org/10.26089/NumMet.v19r216

To the orthogonal expansion theory of the solution to the Cauchy problem for second-order ordinary differential equations

Authors

  • O.B. Arushanyan
  • S.F. Zaletkin

Keywords:

ordinary differential equations
Cauchy problem
approximate analytical methods
numerical methods
orthogonal expansions
shifted Chebyshev series
Markov’s quadrature formulas

Abstract

A solvability theorem is proved for a nonlinear system of equations with respect to the approximate Chebyshev coefficients of the highest derivative in an ordinary differential equation. This theorem is a theoretical substantiation for the previously proposed approximate method of solving canonical systems of second-order ordinary differential equations using orthogonal expansions on the basis of Chebyshev polynomials of the first kind.

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Published

2018-05-02

Issue

Vol. 19 (2018): Issue 2.

Section

Section 1. Numerical methods and applications

Author Biographies

O.B. Arushanyan

Lomonosov Moscow State University
• Head of Laboratory

S.F. Zaletkin

Lomonosov Moscow State University
• Senior Researcher

References

  1. S. F. Zaletkin, “Numerical Integration of Ordinary Differential Equations Using Orthogonal Expansions,” Mat. Model. 22 (1), 69-85 (2010).
  2. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Application of Orthogonal Expansions for Approximate Integration of Ordinary Differential Equations,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 4, 40-43 (2010) [Moscow Univ. Math. Bull. 65 (4), 172-175 (2010)].
  3. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Calculation of Expansion Coefficients of Series in Chebyshev Polynomials for a Solution to a Cauchy Problem,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 5, 24-30 (2012) [Moscow Univ. Math. Bull. 67 (5-6), 211-216 (2012)].
  4. O. B. Arushanyan and S. F. Zaletkin, “Application of Markov’s Quadrature in Orthogonal Expansions,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 6, 18-22 (2009) [Moscow Univ. Math. Bull. 64 (6), 244-248 (2009)].
  5. S. F. Zaletkin, “Markov’s Formula with Two Fixed Nodes for Numerical Integration and Its Application in Orthogonal Expansions,” Vychisl. Metody Programm. 6, 1-17 (2005).
  6. I. S. Berezin and N. P. Zhidkov, Computing Methods (Fizmatgiz, Moscow, 1962; Pergamon, Oxford, 1965).