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Zh. Vychisl. Mat. Mat. Fiz., 2016, Volume 56, Number 11, Pages 1889–1901 (Mi zvmmf10487)  

Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations

A. A. Belolipetskiia, A. M. Ter-Krikorovb

a Dorodnicyn Computing Center, Federal Research Center "Computer Science and Control", Russian Academy of Sciences, Moscow, Russia
b Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow oblast, Russia

Abstract: The functional equation $f(x,\varepsilon)=0$ containing a small parameter $\varepsilon$ and admitting regular and singular degeneracy as $\varepsilon\to0$ is considered. By the methods of small parameter, a function $x_n^0(\varepsilon)$ satisfying this equation within a residual error of $O(\varepsilon^{n+1})$ is found. A modified Newton's sequence starting from the element $x_n^0(\varepsilon)$ is constructed. The existence of the limit of Newton's sequence is based on the NK theorem proven in this work (a new variant of the proof of the Kantorovich theorem substantiating the convergence of Newton's iterative sequence). The deviation of the limit of Newton's sequence from the initial approximation $x_n^0(\varepsilon)$ has the order of $O(\varepsilon^{n+1})$, which proves the asymptotic character of the approximation $x_n^0(\varepsilon)$. The method proposed is implemented in constructing an asymptotic approximation of a system of ordinary differential equations on a finite or infinite time interval with a small parameter multiplying the derivatives, but it can be applied to a wider class of functional equations with a small parameters.

Key words: modified Newton's sequence, small parameters, singular degeneracy, asymptotic approximations, approximate solution to a system of ordinary differential equations, modified Kantorovich theorem.

DOI: https://doi.org/10.7868/S0044466916110053

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English version:
Computational Mathematics and Mathematical Physics, 2016, 56:11, 1859–1871

Bibliographic databases:

UDC: 519.62
Received: 05.10.2015
Revised: 26.02.2016

Citation: A. A. Belolipetskii, A. M. Ter-Krikorov, “Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations”, Zh. Vychisl. Mat. Mat. Fiz., 56:11 (2016), 1889–1901; Comput. Math. Math. Phys., 56:11 (2016), 1859–1871

Citation in format AMSBIB
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\paper Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations
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\yr 2016
\vol 56
\issue 11
\pages 1889--1901
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