Asymptotic solutions of singularly perturbed integro-differential systems with rapidly oscillating coefficients in the case of a simple spectrum

Abdukhafiz Bobodzhanov; Burkhan Kalimbetov; Valeriy Safonov; Abdukhafiz Bobodzhanov; Burkhan Kalimbetov; Valeriy Safonov

doi:10.3934/math.2021512

AIMS Mathematics

2021, Volume 6, Issue 8: 8835-8853. doi: 10.3934/math.2021512

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Research article

Asymptotic solutions of singularly perturbed integro-differential systems with rapidly oscillating coefficients in the case of a simple spectrum

1.
Department of Higher Mathematics, National Research University, MPEI, Krasnokazarmennaya 14, Moscow, 111250, Russia
2.
Department of Mathematics, Khoja Ahmet Yasawi International Kazakh-Turkish University, B. Sattarkhanov 29, Turkestan, 161200, Kazakhstan

Received: 06 January 2021 Accepted: 26 April 2021 Published: 10 June 2021
MSC : 34K26, 45J05

In this paper, we consider a system with rapidly oscillating coefficients, which includes an integral operator with an exponentially varying kernel. The main goal of the work is to develop the algorithm of Lomov's the regularization method for such systems and to identify the influence of the integral term on the asymptotics of the solution of the original problem. The case of identical resonance is considered, i.e. the case when an integer linear combination of the eigenvalues of a rapidly oscillating coefficient coincides with the points of the spectrum of the limit operator is identical on the entire considered time interval. In addition, the case of coincidence of the eigenvalue of a rapidly oscillating coefficient with the points of the spectrum of the limit operator is excluded. This case is supposed to be studied in our subsequent works. More complex cases of resonance (for example, point resonance) require a more thorough analysis and are not considered in this paper.
- singularly perturbed,
- integro-differential equation,
- regularization of an integral,
- space of non-resonant solutions,
- iterative problems,
- solvability of iterative problems
Citation: Abdukhafiz Bobodzhanov, Burkhan Kalimbetov, Valeriy Safonov. Asymptotic solutions of singularly perturbed integro-differential systems with rapidly oscillating coefficients in the case of a simple spectrum[J]. AIMS Mathematics, 2021, 6(8): 8835-8853. doi: 10.3934/math.2021512

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Abstract

In this paper, we consider a system with rapidly oscillating coefficients, which includes an integral operator with an exponentially varying kernel. The main goal of the work is to develop the algorithm of Lomov's the regularization method for such systems and to identify the influence of the integral term on the asymptotics of the solution of the original problem. The case of identical resonance is considered, i.e. the case when an integer linear combination of the eigenvalues of a rapidly oscillating coefficient coincides with the points of the spectrum of the limit operator is identical on the entire considered time interval. In addition, the case of coincidence of the eigenvalue of a rapidly oscillating coefficient with the points of the spectrum of the limit operator is excluded. This case is supposed to be studied in our subsequent works. More complex cases of resonance (for example, point resonance) require a more thorough analysis and are not considered in this paper.

References

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