ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On the Iterative Method of the Study of the Cauchy Problem for a Singularly Perturbed Second-order Linear Differential Equation


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.



  1. 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
  2. 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)
  3. 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)
  4. 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)
  5. 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)
  6. 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)
  7. 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)
  8. Yu. P. Boglaev. An iterative method for the approximate solution of singularly perturbed problems. Soviet Math. Dokl. , 17:543-547, 1976
  9. 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

Full text (pdf)