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 Izv. Akad. Nauk SSSR Ser. Mat., 1970, Volume 34, Issue 4, Pages 827–848 (Mi izv2449)

On the application of linear methods to polynomial approximation of solutions of ordinary differential equations and Hammerstein integral equations

Abstract: Starting from known linear polynomial operators $U_n(\psi;x)$ which generate good approximations to continuous functions $\psi(x)$, the author proposes a method which for a given right-hand side of the equation
$$y'=f(x,y) \tag{1}$$
and given initial conditions enables us to construct polynomials $y_n(x)=y_n(U_n;f;x)$ approximating to the unknown solution of the equation (1) with essentially the same precision as these operators $U_n$ would yield if the solution were given. More precisely, it is shown in this paper that $|y(x)-y_n(U_n;f;x)|\leqslant(1+\alpha_n)\cdot C\|y(x)-U_n(y;x)\|$, $C=\operatorname{const}$, $\alpha_n\downarrow0$, and effective upper bounds are placed on the quantities $C$ and $\alpha_n$. The same procedure is used also for the polynomial approximation of the solutions of $k$-th order equations with $k\geqslant2$, systems of equations, Hamrnerstein integral equations and other integral equations.

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English version:
Mathematics of the USSR-Izvestiya, 1970, 4:4, 835–858

Bibliographic databases:

UDC: 517.9
MSC: 41A10, 41A30, 45B05, 47A58, 34A45

Citation: V. K. Dzyadyk, “On the application of linear methods to polynomial approximation of solutions of ordinary differential equations and Hammerstein integral equations”, Izv. Akad. Nauk SSSR Ser. Mat., 34:4 (1970), 827–848; Math. USSR-Izv., 4:4 (1970), 835–858

Citation in format AMSBIB
\Bibitem{Dzy70} \by V.~K.~Dzyadyk \paper On the application of linear methods to polynomial approximation of solutions of ordinary differential equations and Hammerstein integral equations \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1970 \vol 34 \issue 4 \pages 827--848 \mathnet{http://mi.mathnet.ru/izv2449} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=0279367} \zmath{https://zbmath.org/?q=an:0206.35001|0219.41014} \transl \jour Math. USSR-Izv. \yr 1970 \vol 4 \issue 4 \pages 835--858 \crossref{https://doi.org/10.1070/IM1970v004n04ABEH000935} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. K. Dzyadyk, “The approximation method for approximating solutions of linear differential equations by algebraic polynomials”, Math. USSR-Izv., 8:4 (1974), 937–966
2. V. K. Dzyadyk, L. I. Filozof, “The rate of convergence of Padé approximants for some elementary functions”, Math. USSR-Sb., 35:5 (1979), 615–629
3. N. P. Korneichuk, S. M. Nikol'skii, I. A. Shevchuk, “Vladislav Kirillovich Dzyadyk (on his sixtieth birthday)”, Russian Math. Surveys, 34:4 (1979), 213–221
4. V. K. Dzyadyk, “Asymptotics of diagonal Padé approximants of the functions $\sin z$, $\cos z$, $\operatorname{sinh}z$ and $\operatorname{cosh}z$”, Math. USSR-Sb., 36:2 (1980), 231–249
5. G. A. Dzhanunts, Ya. E. Romm, “The varying piecewise interpolation solution of the Cauchy problem for ordinary differential equations with iterative refinement”, Comput. Math. Math. Phys., 57:10 (2017), 1616–1634
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