Regularization and Construction of Asymptotic Solutions in the Boundary Function Method
DOI:
https://doi.org/10.24160/1993-6982-2019-1-133-138Keywords:
singularly perturbed problem, boundary-layer type asymptotics, regularization, iterative problemsAbstract
In his brief article published in 1977, S.A. Lomov noted that the asymptotics of Vasil'eva—Butuzov’s type boundary layer for a singularly perturbed problem can be obtained by the method of transition to a space of higher dimension or regularization of the original differential operator using a fast independent variable. Unfortunately, a detailed algorithm for constructing such asymptotics is not given in the article, and it remains to be answered whether the asymptotics obtained in it will actually coincide with a boundary-layer type asymptotics. In the present paper, this question is solved positively. However, the regularization itself is divided into two stages: partial regularization and complete regularization. In the course of partial regularization, it is not possible to obtain an "extended" problem the iterative systems of which admit solutions in the form of the sum of two functions with separated independent variables (one of which is an ordinary variable, and the other is a regularizing variable) The complete regularization has at its heart the procedure for constructing an "extended" system, which was applied by the authors in their papers devoted to regularization of integral operators, the essence of which consists in regularizing the original problem in the class of formal asymptotic series in powers of a small parameter with coefficients in the form of the sum of functions with separated variables. The asymptotics obtained using such procedure coincides completely with the boundary-layer type asymptotics, a result that confirms S.A. Lomov’s hypothesis.
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Для цитирования: Бободжанов А.А., Сафонов В.Ф. Регуляризация и построение асимптотических решений в методе пограничных функций // Вестник МЭИ. 2019. № 1. С. 133—138. DOI: 10.24160/1993-6982-2019-1-133-138.
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For citation: Bobodzhanov A.A., Safonov V.F. Regularization and Construction of Asymptotic Solutions in the Boundary Function Method. MPEI Vestnik. 2019;1:133—138. (in Russian). DOI: 10.24160/1993-6982-2019-1-133-138.
