Abstract:
This paper is concerned with a singularly perturbed integral system with rapidly varying kernels. In contrast to integral equations with slowly varying kernels, where regularization is carried out with respect to the spectrum of the “diagonal” kernel, the regularization problem in the case of rapidly varying kernels is solved in terms of the spectrum of a certain extended operator. The goal here is to construct this operator and develop an appropriate algorithm for asymptotic solutions.
Citation:
A. A. Bobodzhanov, V. F. Safonov, “Volterra integral equations with rapidly varying kernels and their asymptotic integration”, Sb. Math., 192:8 (2001), 1139–1164
\Bibitem{BobSaf01}
\by A.~A.~Bobodzhanov, V.~F.~Safonov
\paper Volterra integral equations with rapidly varying kernels and their asymptotic integration
\jour Sb. Math.
\yr 2001
\vol 192
\issue 8
\pages 1139--1164
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\crossref{https://doi.org/10.1070/SM2001v192n08ABEH000586}
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Linking options:
https://www.mathnet.ru/eng/sm586
https://doi.org/10.1070/SM2001v192n08ABEH000586
https://www.mathnet.ru/eng/sm/v192/i8/p53
This publication is cited in the following 24 articles:
M. A. Bobodzhanova, B. T. Kalimbetov, V. F. Safonov, “Regularised asymptotic solutions of integro-differential equations with zero operator of the differential part and with rapidly oscillating inhomogeneity”, Applicable Analysis, 2025, 1
A. A Bobodzhanov, B. T Kalimbetov, V. F Safonov, “Singulyarno vozmushchennye integro-differentsial'nye sistemy s yadrami, zavisyashchimi ot resheniy differentsial'nykh uravneniy”, Differencialʹnye uravneniâ, 59:5 (2023), 693
Burkhan Kalimbetov, Valeriy Safonov, Dinara Zhaidakbayeva, “Asymptotic Solution of a Singularly Perturbed Integro-Differential Equation with Exponential Inhomogeneity”, Axioms, 12:3 (2023), 241
A. A. Bobodzhanov, B. T. Kalimbetov, V. F. Safonov, “Singularly Perturbed Integro-Differential Systems with Kernels Depending on Solutions of Differential Equations”, Diff Equat, 59:5 (2023), 707
Kalimbetov B.T. Tuychiev O.D., “Asymptotic Solution of the Cauchy Problem For the Singularly Perturbed Partial Integro-Differential Equation With Rapidly Oscillating Coefficients and With Rapidly Oscillating Heterogeneity”, Open Math., 19 (2021), 244–258
Burkhan Kalimbetov, Valery Safonov, “Singularly Perturbed Integro-Differential Equations With Rapidly Oscillating Coefficients and With Rapidly Changing Kernel in the Case of a Multiple Spectrum”, WSEAS TRANSACTIONS ON MATHEMATICS, 20 (2021), 84
Bobodzhanova M.A., Safonov V.F., “Regularized Asymptotics of Solutions of Integro-Differential Equations With Zero Operator in the Differential Part and With Slowly and Rapidly Varying Kernels”, Differ. Equ., 56:4 (2020), 523–532
Bobodzhanov A., Safonov V., Kachalov V., “Asymptotic and Pseudoholomorphic Solutions of Singularly Perturbed Differential and Integral Equations in the Lomov'S Regularization Method”, Axioms, 8:1 (2019), 27
Kalimbetov B.T. Safonov V.F., “Integro-Differentiated Singularly Perturbed Equations With Fast Oscillating Coefficients”, Bull. Karaganda Univ-Math., 94:2 (2019), 33–47
Bobodzhanov A.A. Kalimbetov B.T. Safonov V.F., “Singularly Perturbed Control Problems in the Case of the Stability of the Spectrum of the Matrix of An Optimal System”, Bull. Karaganda Univ-Math., 96:4 (2019), 22–38
M. A. Bobodzhanova, V. F. Safonov, “Regulyarizovannye asimptoticheskie resheniya integrodifferentsialnykh uravnenii s nulevym operatorom differentsialnoi chasti i s neskolkimi bystro izmenyayuschimisya yadrami”, Sib. elektron. matem. izv., 15 (2018), 1566–1575
A. A. Bobodzhanov, V. F. Safonov, “Asymptotic solutions of Fredholm integro-differential equations with rapidly changing kernels and irreversible limit operator”, Russian Math. (Iz. VUZ), 59:10 (2015), 1–15
A. A. Bobodzhanov, V. F. Safonov, “The method of normal forms for singularly perturbed systems of Fredholm integro-differential equations with rapidly varying kernels”, Sb. Math., 204:7 (2013), 979–1002
Safonov V.F., Bobodzhanov A.A., “Razvitie idei S.A. Lomova v teorii singulyarno vozmuschennykh integralnykh i integrodifferentsialnykh uravnenii. k 90-letiyu so dnya rozhdeniya S.A. Lomova”, Vestnik moskovskogo energeticheskogo instituta, 2012, no. 6, 5–12
Bobodzhanova M.A., Safonov V.F., “Asymptotic analysis of singularly perturbed integro-differential equations with zero operator in the differential part”, Differ Equ, 47:4 (2011), 516–533
A. A. Bobodzhanov, V. F. Safonov, ““Splashes” in Fredholm Integro-Differential Equations with Rapidly Varying Kernels”, Math. Notes, 85:2 (2009), 153–167
Bobodzhanov A.A., Safonov V.F., “Regularized asymptotics of solutions of systems of Fredholm integro-differential equations with rapidly varying kernels”, Differ. Equ., 45:2 (2009), 226–239
Citation in format AMSBIB
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