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Asymptotic and Pseudoholomorphic Solutions of Singularly Perturbed Differential and Integral Equations in the Lomov’s Regularization Method

National Research University “MPEI”, 111250 Moscow, Russia
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Author to whom correspondence should be addressed.
Sections devoted to singularly perturbed integral equations were written by Bobodzhanov A.A. and Safonov V.F., the sections connected with holomorphic regularization were written by Kachalov V.I.
Current address: National Research University “MPEI”, Ul. Krasnokazarmennaya 14, 111250 Moscow, Russia; Tel.: +8-495-362-71-31.
Axioms 2019, 8(1), 27; https://doi.org/10.3390/axioms8010027
Received: 9 December 2018 / Revised: 2 February 2019 / Accepted: 21 February 2019 / Published: 1 March 2019
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Abstract

We consider a singularly perturbed integral equation with weakly and rapidly varying kernels. The work is a continuation of the studies carried out previously, but these were focused solely on rapidly changing kernels. A generalization for the case of two kernels, one of which is weakly, and the other rapidly varying, has not previously been carried out. The aim of this study is to investigate the effects introduced into the asymptotics of the solution of the problem by a weakly varying integral kernel. In the second part of the work, the problem of constructing exact (more precise, pseudo-analytic) solutions of singularly perturbed problems is considered on the basis of the method of holomorphic regularization developed by one of the authors of this paper. The power series obtained with the help of this method for the solutions of singularly perturbed problems (in contrast to the asymptotic series constructed in the first part of this paper) converge in the usual sense. View Full-Text
Keywords: singularly perturbed; integral equations; regularization of the integral; weakly and rapidly changing kernel; holomorphic integrals; family of homomorphisms; asymptotic and pseudoholomorphic solutions singularly perturbed; integral equations; regularization of the integral; weakly and rapidly changing kernel; holomorphic integrals; family of homomorphisms; asymptotic and pseudoholomorphic solutions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Bobodzhanov, A.; Safonov, V.; Kachalov, V. Asymptotic and Pseudoholomorphic Solutions of Singularly Perturbed Differential and Integral Equations in the Lomov’s Regularization Method. Axioms 2019, 8, 27.

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