Geodesic
Approximate integration of ordinary differential equations using Chebyshev series with precision control
Matematičeskoe modelirovanie, Tome 34 (2022) no. 6, pp. 53-74.

Voir la notice de l'article provenant de la source Math-Net.Ru

An approximate method of solving the Cauchy problem for canonical systems of second order ordinary differential equations is considered. The method is based on using the shifted Chebyshev series and a Markov quadrature formula. Some approaches are given to estimate the errors of an approximate solution and its derivative expressed by partial sums of a certain order shifted Chebyshev series. The errors are estimated using the second approximation of the solution calculated in a special way and expressed by a partial sum of a higher order series. An algorithm of partitioning the integration interval into elementary subintervals to ensure the computation of the solution and its derivative with prescribed accuracy is discussed on the basis of proposed approaches to error estimation.
Keywords: ordinary differential equations, approximate analytical methods, numerical methods, orthogonal expansions, shifted Chebyshev series, polynominal approximation, precision control, error estimate, automatic step size control.
Mots-clés : Markov quadrature formulas 
@article{MM_2022_34_6_a3,
     author = {S. F. Zaletkin},
     title = {Approximate integration of ordinary differential equations using {Chebyshev} series with precision control},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {53--74},
     publisher = {mathdoc},
     volume = {34},
     number = {6},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2022_34_6_a3/}
}
TY  - JOUR
AU  - S. F. Zaletkin
TI  - Approximate integration of ordinary differential equations using Chebyshev series with precision control
JO  - Matematičeskoe modelirovanie
PY  - 2022
SP  - 53
EP  - 74
VL  - 34
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2022_34_6_a3/
LA  - ru
ID  - MM_2022_34_6_a3
ER  - 
%0 Journal Article
%A S. F. Zaletkin
%T Approximate integration of ordinary differential equations using Chebyshev series with precision control
%J Matematičeskoe modelirovanie
%D 2022
%P 53-74
%V 34
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2022_34_6_a3/
%G ru
%F MM_2022_34_6_a3
S. F. Zaletkin. Approximate integration of ordinary differential equations using Chebyshev series with precision control. Matematičeskoe modelirovanie, Tome 34 (2022) no. 6, pp. 53-74. http://geodesic.mathdoc.fr/item/MM_2022_34_6_a3/

[1] S. F. Zaletkin, “Chislennoe integrirovanie obyknovennykh differentsial'nykh uravnenij s ispol'zovaniem ortogonal'nykh razlozhenij”, Mat. mod., 22:1 (2010), 69–85 | MR | Zbl

[2] O. B. Arushanyan, N. I. Volchenskova, S. F. Zaletkin, “Application of Orthogonal Expansions for Approximate Integration of Ordinary Differential Equations”, Moscow University Mathematics Bulletin, 65:4 (2010), 172–175 | DOI | MR | MR | Zbl

[3] O. B. Arushanyan, N. I. Volchenskova, S. F. Zaletkin, “Calculation of Expansion Coefficients of Series in Chebyshev Polynomials for a Solution to a Cauchy Problem”, Moscow University Mathematics Bulletin, 67:5/6 (2012), 211–216 | DOI | MR | Zbl

[4] O. B. Arushanyan, S. F. Zaletkin, “K teorii vychisleniia ortogonal'nogo razlozheniia resheniia zadachi Koshi dlia obyknovennykh differentsial'nykh uravnenij vtorogo poriadka”, Vychislitel'nye metody i programmirovanie, 19 (2018), 178–184

[5] O. B. Arushanyan, S. F. Zaletkin, “Ob odnoj realizatsii metoda riadov Chebyshova dlia priblizhennogo analiticheskogo resheniia system obyknovennykh differentsial'nykh uravnenij vtorogo poriadka”, Vychislitel'nye metody i programmirovanie, 20 (2019), 210–216

[6] O. B. Arushanyan, S. F. Zaletkin, “Application of Markov's Quadrature in Orthogonal Expansions”, Moscow University Mathematics Bulletin, 64:6 (2009), 244–248 | DOI | MR | Zbl

[7] S. F. Zaletkin, “Formula chislennogo integrirovaniia Markova s dvumia fiksirovannymi uzlami i ee primenenie v ortogonal'nykh razlozheniiakh”, Vychislitel'nye metody i programmirovanie, 6 (2005), 1–17

[8] O. B. Arushanyan, S. F. Zaletkin, “Ob otsenke pogreshnosti priblizhennogo resheniia obyknovennykh differentsial'nykh uravnenij, opredelennogo s pomoshch'iu riadov Chebyshova”, Vychislitel'nye metody i programmirovanie, 21 (2020), 241–250

[9] N. N. Kalitkin, P. V. Koryakin, Chislennye metody, Uchebnik, v 2 kn., v. 2, Metody matematicheskoy fiziki, Izd. tsentr «Akademia», 2013, 304 pp.

[10] E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations. Nonstiff Problems, Springer-Verlag, Berlin–Heidelberg–New York–London–Paris–Tokyo, 1987 | MR | MR | Zbl

[11] N. S. Bakhvalov, N. P. Zhidkov, G. M. Kobel'kov, Chislennye metody, Nauka, M., 1987, 600 pp. | MR

[12] O. B. Arushanyan, S. F. Zaletkin, Chislennoe reshenie obyknovennykh differentsial'nykh uravnenij na Fortrane, Izd-vo MGU, M., 1990, 336 pp. | MR