An implementation of the Chebyshev series method for the approximate analytical solution of second-order ordinary differential equations
DOI:
https://doi.org/10.26089/NumMet.v20r210Keywords:
ordinary differential equations of second order, canonical systems of ordinary differential equations, approximate analytical methods, numerical methods, orthogonal expansions, shifted Chebyshev series, Markov quadrature formulas, polynomial approximationAbstract
In this paper we consider a stochastic model of the galactic dynamo in which the coefficient of turbulent diffusion is considered as a random process with renewal. The numerical simulation of statistical moments as well as two-point and three-point correlators of the magnetic field showing the relation between its values at various time instants is performed. The presence of intermittency expressed in the progressive growth of moments and correlators in the case of "quiet" regions of galaxies with a small fraction of the ionized hydrogen component is shown. The numerical results are compared with the results obtained analytically earlier.
References
- 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)].
- 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).
- S. F. Zaletkin, “Numerical Integration of Ordinary Differential Equations Using Orthogonal Expansions,” Mat. Model. 22 (1), 69-85 (2010).
- 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)].
- 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)].
- O. B. Arushanyan and S. F. Zaletkin, “Justification of an Approach to Application of Orthogonal Expansions for Approximate Integration of Canonical Systems of Second Order Ordinary Differential Equations,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 3, 29-33 (2018) [Moscow Univ. Math. Bull. 73 (3), 111-115 (2018)].
- O. B. Arushanyan and S. F. Zaletkin, “To the Orthogonal Expansion Theory of the Solution to the Cauchy Problem for Second-Order Ordinary Differential Equations,” Vychisl. Metody Programm. 19, 178-184 (2018).
- 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. 14 (4), 59-68 (2009).
- 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).
- 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).
- C. Lanczos, Applied Analysis (Prentice-Hall, Englewood Cliffs, 1956; Fizmatgiz, Moscow, 1961).
- R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1962; Nauka, Moscow, 1972).
- S. Paszkowski, Numerical Applications of Chebyshev Polynomials and Series (PWN, Warsaw, 1975 [in Polish]; Nauka, Moscow, 1983).
Downloads
Published
25-03-2019
How to Cite
Арушанян О., Залеткин С. An Implementation of the Chebyshev Series Method for the Approximate Analytical Solution of Second-Order Ordinary Differential Equations // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2019. 20. 97-103. doi 10.26089/NumMet.v20r210
Issue
Section
Section 1. Numerical methods and applications
