WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 20, 2021
Singularly Perturbed Integro-Differential Equations with Rapidly Oscillating Coefficients and with Rapidly Changing Kernel in the Case of a Multiple Spectrum
Authors: ,
Abstract: 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.
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Keywords: singular perturbation, integro-differential equation, rapidly oscillating coefficient,
regularization, asymptotic convergence
Pages: 84-96
DOI: 10.37394/23206.2021.20.9