On the Iterative Method of the Study of the Cauchy Problem for a Singularly Perturbed Second-order Linear Differential Equation
Author(s):
Evgeny Evgen'evich Bukzhalev
M. V. Lomonosov Moscow State University
Department of Mathematics
Faculty of Physics
associate professor, PhD in physics and mathematics
bukzhalev@mail.ru
Abstract:
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.
Keywords
- Banach fixed-point theorem
- method of asymptotic iterations
- method of boundary functions
- method of regularization of singular perturbations
- singular perturbations
References:
- 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
