Mathematical Modelling, Numerical Methods and Software Systems

Asymptotic expansions of solutions of singularly perturbed equations

V. I. Uskov

Voronezh State University of Forestry and Technologies named after G.F. Morozov, Voronezh

Abstract: We consider a first-order equation in a Banach space with a small parameter at the derivative and a second-order perturbation of smallness on the right-hand side. A solution to the Cauchy problem is constructed in the form of an asymptotic expansion in powers of a small parameter by the Vasilieva-Vishik-Lyusternik method. The operator A on the right-hand side is degenerate: we consider the case of possessing the property of having a number 0 by a normal eigenvalue and a two-dimensional kernel; core elements have no attached. Formulas for calculating the components of the regular and boundary layer parts of the expansion are determined. A condition for the regularity of degeneration is obtained. The expansion is shown to be asymptotic. An illustrative example is given.

Keywords: first-order equation in a Banach space, small parameter at the highest derivative, perturbation square on the right-hand side, closed operator, 0-normal eigenvalue, asymptotics, Vasil'eva-Vishik-Lyusternik method.

UDC: 517.928

PACS: 02.30.Hq

MSC: 34E15

Received: 21.11.2021
Revised: 13.01.2022

DOI: 10.26456/vtpmk629

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