DOI: https://doi.org/10.26089/NumMet.v16r223

On an approximate analytical method of solving ordinary differential equations

Authors

  • O.B. Arushanyan
  • N.I. Volchenskova
  • S.F. Zaletkin

Keywords:

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

Abstract

The application of shifted Chebyshev series for solving ordinary differential equations is described. This approach is based on the approximation of the solution to the Cauchy problem for a normal system of ordinary differential equations and its derivatives by partial sums of Fourier series in the Chebyshev polynomials of the first kind. The coefficients of the series are determined by an iterative process with the use of Markov’s quadrature formulas. The approximation properties of shifted Chebyshev series allow us to propose an approximate analytical method for ordinary differential equations. A number of examples are considered to illustrate the application of partial sums of Chebyshev series for approximate representations of the solutions to the Cauchy problems for ordinary differential equations.

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Published

2015-05-06

Issue

Vol. 16 (2015): Issue 2.

Section

Section 1. Numerical methods and applications

Author Biographies

O.B. Arushanyan

Lomonosov Moscow State University,
Research Computing Center
• Head of Laboratory

N.I. Volchenskova

Lomonosov Moscow State University,
Research Computing Center
• Senior Researcher

S.F. Zaletkin

Lomonosov Moscow State University,
Research Computing Center
• Senior Researcher

References

  1. K. I. Babenko, Fundamentals of Numerical Analysis (Nauka, Moscow, 1986) [in Russian].
  2. I. P. Mysovskikh, Lectures on Numerical Methods (St. Petersburg Univ., St. Petersburg, 1998) [in Russian].
  3. V. P. II’in and Yu. A. Kuznetsov, Algebraic Foundations of Numerical Analysis (Nauka, Novosibirsk, 1986) [in Russian].
  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. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “On Some Properties of Partial Sums for Chebyshev Series,” Differen. Uravn. Protsessy Upravl. 13 (3), 26-34 (2009).
  7. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Approximate Integration of Ordinary Differential Equations on the Basis of Orthogonal Expansions,” Differen. Uravn. Protsessy Upravl. 13 (4), 59-68 (2009).
  8. S. F. Zaletkin, “Numerical Integration of Ordinary Differential Equations Using Orthogonal Expansions,” Mat. Model. 22 (1), 69-85 (2010).
  9. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Approximate Solution of Ordinary Differential Equations Using Chebyshev Series,” Sib. Elektron. Mat. Izv. 7, 122-131 (2010).
  10. 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)].
  11. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “On Calculation of Chebyshev Series Coefficients for the Solutions to Ordinary Differential Equations,” Sib. Elektron. Mat. Izv. 8, 273-283 (2011).
  12. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Calculation of the Coefficients of Orthogonal Expansions for the Solutions to Ordinary Differential Equations,” Differen. Uravn. Protsessy Upravl. 15 (2), 41-47 (2011).
  13. 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)].
  14. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “A Method of Solving the Cauchy Problem for Ordinary Differential Equations Using Chebyshev Series,” Vychisl. Metody Programm. 14, 203-214 (2013).
  15. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Application of Chebyshev Series for the Integration of Ordinary Differential Equations,” Sib. Elektron. Mat. Izv. 11, 517-531 (2014).
  16. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin , “An Approximate Method for Integration of Ordinary Differential Equations,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 6, 43-46 (2013) [Moscow Univ. Math. Bull. 68 (6), 292-294 (2013)].
  17. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “On an Approach to Integration of Ordinary Differential Equations with the Use of Series,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 6, 57-60 (2014) [Moscow Univ. Math. Bull. 69 (6), 272-274 (2014)].