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 Mat. Sb., 2005, Volume 196, Number 2, Pages 29–56 (Mi msb1265)

Singularly perturbed integro-differential systems with contrast structures

A. A. Bobodzhanov, V. F. Safonov

Moscow Power Engineering Institute (Technical University)

Abstract: An integro-differential system with an eigenvalue of the limiting operator of the differential part taking the value 0 is considered. An algorithm is developed allowing one to obtain asymptotic solutions (of an arbitrary order) by the method of normal forms. Contrast structures (internal transition layers) in solutions of the problem under consideration are investigated on the basis of the analysis of the leading term of the asymptotic solution. Contrast structures are shown to result from the instability of the spectrum of the limiting operator and the presence of an inhomogeneity. The role of the kernel of the integral operator in the development of contrast structures is also cleared up. In integral systems with diagonal degeneration of the kernel $(K(t,t)\equiv0)$ the integral term plays no role in the development of contrast structures and, conversely, if the kernel is non-degenerate, then it is significant for the development of contrast structures.

DOI: https://doi.org/10.4213/sm1265

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English version:
Sbornik: Mathematics, 2005, 196:2, 173–200

Bibliographic databases:

UDC: 517.968
MSC: Primary 35J05; Secondary 35Mxx, 34E15

Citation: A. A. Bobodzhanov, V. F. Safonov, “Singularly perturbed integro-differential systems with contrast structures”, Mat. Sb., 196:2 (2005), 29–56; Sb. Math., 196:2 (2005), 173–200

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb1265
• https://doi.org/10.4213/sm1265
• http://mi.mathnet.ru/eng/msb/v196/i2/p29

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Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. A. Bobodzhanov, V. F. Safonov, “Asymptotic analysis of integro-differential systems with an unstable spectral value of the integral operator's kernel”, Comput. Math. Math. Phys., 47:1 (2007), 65–79
2. A. A. Bobodzhanov, V. F. Safonov, “Regularized Asymptotics of Solutions of Systems of Fredholm Integro-Differential Equations with Rapidly Varying Kernels”, Differ. Equ., 45:2 (2009), 226–239
3. Bobodzhanov A.A., Safonov V.F., “Sistema integrodifferentsialnykh uravnenii Fredgolma s bystro izmenyayuschimisya yadrami v sluchae diskretnoi neobratimosti predelnogo operatora”, Vestn. Mosk. energeticheskogo in-ta, 2009, no. 6, 62–71
4. Bobodzhanova M.A., “Obosnovanie algoritma metoda realizatsii dlya nelineinykh integrodifferentsialnykh uravnenii s nulevym operatorom differentsialnoi chasti”, Vestnik moskovskogo energeticheskogo instituta, 2011, no. 6, 85–95
5. 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
6. 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
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