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JOURNALS // Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports] // Archive

Sib. Èlektron. Mat. Izv., 2024 Volume 21, Issue 1, Pages 453–462 (Mi semr1695)

Differentical equations, dynamical systems and optimal control

Regularized asymptotic solutions of integro-differential equations with fast and slow variables

V. S. Besov

Moscow Power Engineering Institute, Krasnokazarmennaya Ulitsa, 14, 111250, Moscow, Russia

Abstract: The paper considers a nonlinear integro-differential system with fast and slow variables. Such systems have not been considered previously from the point of view of constructing regularized (according to Lomov) asymptotic solutions. Known works were mainly devoted to the construction of the asymptotics of the Butuzov-Vasil'eva boundary layer type, which, as is known, can be applied only if the spectrum of the matrix of the first variation (on the degenerate solution) is located strictly in the open left half-plane of a complex variable. In the case when the spectrum of the indicated matrix falls on the imaginary axis, the method of regularization by S.A. Lomov. However, this method was developed mainly for singularly perturbed differential systems that do not contain integral terms, or for integro-differential problems without slow variables. In this paper, the regularization method is generalized to two-dimensional integro-differential equations with fast and slow variables.

Keywords: nonlinear systems, integro-differential equations, regularization, singular perturbations, fast variables, slow variables.

UDC: 517.95

MSC: 32A40, 32A55, 32S99

Received March 26, 2024, published July 18, 2024

DOI: doi.org/10.33048/semi.2024.21.032



© Steklov Math. Inst. of RAS, 2025