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 Daghestan Electronic Mathematical Reports, 2017, Issue 7, Pages 66–76 (Mi demr39)

Approximation of the solution of the Cauchy problem for nonlinear ODE systems by means of Fourier series in functions orthogonal in the sense of Sobolev

I. I. Sharapudinovab

a Daghestan Scientific Centre of Russian Academy of Sciences, Makhachkala

Abstract: Consider the systems of functions ${\varphi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ orthonormal with respect to a Sobolev-type inner product of the form $\langle f,g\rangle= \sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+\int_{a}^{b}f^{(r)}(x)g^{(r)}\rho(x)(x)dx$ generated by a given orthonormal system of functions ${\varphi}_{n}(x)$ $( n=0,1,\ldots)$. It is shown that the Fourier series in the system ${\varphi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ and their partial sums are a convenient and very effective tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).

Keywords: the Cauchy problem, Fourier series, Sobolev orthogonal functions

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00486a This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 16-01-00486a)

DOI: https://doi.org/10.31029/demr.7.8

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UDC: 517.538
Revised: 18.05.2017
Accepted:19.05.2017

Citation: I. I. Sharapudinov, “Approximation of the solution of the Cauchy problem for nonlinear ODE systems by means of Fourier series in functions orthogonal in the sense of Sobolev”, Daghestan Electronic Mathematical Reports, 2017, no. 7, 66–76

Citation in format AMSBIB
\Bibitem{Sha17} \by I.~I.~Sharapudinov \paper Approximation of the solution of the Cauchy problem for nonlinear ODE systems by means of Fourier series in functions orthogonal in the sense of Sobolev \jour Daghestan Electronic Mathematical Reports \yr 2017 \issue 7 \pages 66--76 \mathnet{http://mi.mathnet.ru/demr39} \crossref{https://doi.org/10.31029/demr.7.8} 

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This publication is cited in the following articles:
1. M. S. Sultanakhmedov, “Cauchy problem for the difference equation and Sobolev orthogonal functions on the finite grid, generated by discrete orthogonal functions”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 7, 77–85
2. I. I. Sharapudinov, M. G. Magomed-Kasumov, “Chislennyi metod resheniya zadachi Koshi dlya sistem obyknovennykh differentsialnykh uravnenii s pomoschyu ortogonalnoi v smysle Soboleva sistemy, porozhdennoi sistemoi kosinusov”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 8, 53–60
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