Moscow University Mathematics Bulletin

, Volume 74, Issue 3, pp 127–130

# On Some Analytic Method for Approximate Solution of Systems of Second Order Ordinary Differential Equations

• O. B. Arushanyan
• S. F. Zaletkin
Article

## Abstract

An approach to using Chebyshev series to solve canonical second-order ordinary differential equations is described. This approach is based on the approximation of the solution to the Cauchy problem and its first and second derivatives by partial sums of shifted Chebyshev series. The coefficients of the series are determined by an iterative process using the Markov quadrature formula. It is shown that the described approach allows one to propose an approximate analytical method of solving the Cauchy problem. A number of canonical second-order ordinary differential equations are considered to represent their approximate analytical solutions in the form of partial sums of shifted Chebyshev series.

## References

1. 1.
S. F. Zaletkin, “Numerical Integration of Ordinary Differential Equations Using Orthogonal Expansions,” Matem. Model. 22 (1), 69 (2010).
2. 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., Matem. Mekhan., No. 4, 40 (2010) [Moscow Univ. Math. Bull. 65 (4), 172 (2010)].Google Scholar
3. 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. Matem. Mekhan., No. 5, 24 (2012).Google Scholar
4. 4.
O. B Arushanyan and S. F. Zaletkin, “Application of Markov’s Quadrature in Orthogonal Expansions,” Vestn. Mosk. Univ., Matem. Mekhan., No. 6, 18 (2009) [Moscow Univ. Math. Bull. 64 (6), 244 (2009)].Google Scholar
5. 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 (2005).Google Scholar
6. 6.
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., Matem. Mekhan., No. 3, 29 (2018)Google Scholar
7. 7.
O. B. Arushanyan and S. F. Zaletkin, “To the Theory of of Calculation of Orthogonal Expansion for Solution to Cauchy Problem for Second Order Ordinary Differential Equations,” Vychisl. Metody Programm. 19, 178 (2018).Google Scholar
8. 8.
O. B. Arushanyan, N. I. Volchenskova and S. F. Zaletkin, “Approximate Integration of Ordinary Differential Equations on the Base of Orthogonal Expansions,” Differ. Uravn. Processy Upravl. 14 (4), 59 (2009).
9. 9.
O. B. Arushanyan, N. I. Volchenskova and S. F. Zaletkin, “Approximate Solution of Ordinary Differential Equations Using Chebyshev Series,” Sib. Elektron. Matem. Izv., 7, 122 (2010).
10. 10.
O. B. Arushanyan, N. I. Volchenskova and S. F. Zaletkin, “Calculation of Coefficients of Chebyshev Series for Solution of Ordinary Differential Equations,” Sib. Elektron. Matem. Izv., 8, 273 (2011).