Singularly perturbed integro-differential equations with rapidly oscillating coefficients and with rapidly changing kernel in the case of a multiple spectrum

Kalimbetov B. Safonov V.
2021 World Scientific and Engineering Academy and Society

WSEAS Transactions on Mathematics
2021 #20 84 - 96 pp.

The paper investigates a system with rapidly oscillating coefficients and with a rapidly decreasing kernel of the integral operator. Previously, only differential problems of this type were studied in which the integral term was absent. The presence of an integral operator significantly affects the development of an algorithm for asymptotic solutions, for the implementation of which it is necessary to take into account essentially singularities generated by the rapidly decreasing spectral value of the kernel of the integral operator. In addition, resonances can arise in the problem under consideration (i.e., the case can be realized when an integer linear combination of the eigenvalues of the rapidly oscillating coefficient coincides with the points of the spectrum of the limiting operator over the entire considered time interval), as well as the case where the eigenvalue of the rapidly oscillating coefficient coincides with the points spectrum of the limiting operator. This case generates a multiple spectrum of the original singularly perturbed integro-differential system. A similar problem was previously considered in the case of a simple spectrum. More complex cases of resonance (for example, point resonance) require more careful analysis and are not considered in this article.

Asymptotic convergence , Integro-differential equation , Rapidly oscillating coefficient , Regularization , Singular perturbation

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Ahmet Yassawi University Natural Sciences Institute, Sattarkhanov ave. 29, Turkestan, 161200, Kazakhstan
National Research University, Moscow Power Engineering Institute, Krasnokazarmennaya Street 14, Moscow, 111250, Russian Federation

Ahmet Yassawi University Natural Sciences Institute
National Research University

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