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Русская версия

**Evgeny Evgen'evich Bukzhalev**

M. V. Lomonosov Moscow State University

Department of Mathematics

Faculty of Physics

associate professor, PhD in physics and mathematics

We construct a sequence that converges both in the asymptotic and usual sense (with respect to the norm of the space of continuous functions) to the solution of the Cauchy problem for a singularly perturbed second-order linear homogeneous differential equation. The similar sequence was constructed for a first-order linear homogeneous equation as well. Using this equation as an example we demonstrate the justification of the asymptotics obtained by the method of boundary functions.

- Banach fixed-point theorem
- method of asymptotic iterations
- method of boundary functions
- method of regularization of singular perturbations
- singular perturbations

- A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov.
*Differential Equations*. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, Heidelberg, 1985. 240 pp - A. A. Barashkov, V. A. Borkhalenko. Limits of applicability of an iterative ‑ asymptotic method of the solution of the inverse problems for periodic structures.
*MPEI Vestnik*, (6):141-146, 2013. (in Russian) - N. D. Kopachevskii and V. P. Smolich,
*Vvedeniye v asimptoticheskiye metody: Spetsial'nyy kurs lektsiy*[Introduction to Asymptotic Methods, Special Course of Lectures]. Simferopol’, Tavrich. Nats. Univ. Publ., 2009. 52 p. (in Russian) - A. B. Vasil’eva and V. F. Butuzov,
*Asimptoticheskiye razlozheniya resheniy singulyarno vozmushchennykh uravneniy*[Asymptotic Expansions of Solutions to Singularly Perturbed Equations]. Moscow, Nauka Publ., 1973. 272 p. (in Russian) - A. B. Vasil’eva and V. F. Butuzov,
*Asimptoticheskiye metody v teorii singulyarnykh vozmushcheniy*[Asymptotic Methods in the Theory of Singular Perturbations]. Moscow, Vysshaya Shkola Publ., 1990. 208 p. (in Russian) - S. A. Lomov,
*Vvedeniye v obshchuyu teoriyu singulyarnykh vozmushcheniy*[Introduction to the General Theory of Singular Perturbations]. Moscow, Nauka Publ., 1981. 400 p. (in Russian) - S. A. Lomov, I. S. Lomov.
*Osnovy matematicheskoy teorii pogranichnogo sloya*[Fundamentals of the mathematical theory of the boundary layer]. Moscow, Publishing house of MSU, 2011. 456 p. (in Russian) - Yu. P. Boglaev. An iterative method for the approximate solution of singularly perturbed problems.
*Soviet Math. Dokl.*, 17:543-547, 1976 - Yu. P. Boglaev, A. V. Zhdanov, and V. G. Stel’makh. Uniform approximations for solutions of certain singularly perturbed nonlinear equations.
*Differ. Equations*, 14:273-281, 1978