Regularization of Integral Operators in Singularly Perturbed Problems Using Normal Forms
Abstract
The main difficulty encountered in generalizing S.А. Lomov’s regularization method for integral and integro-differential singularly perturbed problems lies in bringing integral operators to a regular kind. If there are no integral operators, that is, if there is only a differential operator, the corresponding singularly perturbed problem is usually regularized by the spectrum of the limit operator using the well-known formula of complex differentiation. For integral operators, there is no an analog of the complex differentiation formula; therefore, it is not possible to directly construct an extension of the integral operator. In one of the authors' works that was addressed to generalization of S.A. Lomov’s ideas, a regularization method for differential systems with the use of normal forms was developed. With such regularization, additional (regularizing) variables are not entered directly from the limit operator spectrum, but are indirectly calculated using some normal differential form, the solution of which is written in quadratures. In the case of nonzero points of the limit operator’s simple spectrum, such regularization coincides with the regularization along the spectrum developed by S.А. Lomov. However, if there are eigenvalues of the limit operator that vanish in at least one point of the considered time interval, S.А. Lomov’s classical regularization does not make it possible to construct an extension of an integral operator and carry out its full regularization. This difficulty has been fully eliminated by applying regularization with the use of normal forms.
The basic ideas of the method of normal forms have been demonstrated taking as an example the simplest integro-differential problem with spectrum instability on a continual set, and the statement using which integral operators are regularized is given. The main term of the problem solution asymptotics is written, after studying of which a conclusion is drawn about the presence of two layers in the solution: the boundary layer in the neighborhood of the initial point of the considered time interval and the inner transition layer (a contrast structure) in the neighborhood of the instability set end point.
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Для цитирования: Бободжанов А.А., Сафонов В.Ф. Регуляризация интегральных операторов в сингулярно возмущенных задачах с помощью нормальных форм / Вестник МЭИ. 2019. № 6. С. 131—137. DOI: 10.24160/1993-6982-2019-6-131-137.
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For citation: Bobodzhanov A.A.. Safonov V.F. Regularization of Integral Operators in Singularly Perturbed Problems Using Normal Forms. Bulletin of MPEI. 2019;6:131—137. (in Russian). DOI: 10.24160/1993-6982-2019-6-131-137.
