This article is cited in 2 papers

Scientific articles

Asymptotic solution of the Cauchy problem for the first-order equation with perturbed Fredholm operator

V. I. Uskov

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

Abstract: We consider the Cauchy problem for a first-order differential equation in a Banach space. The equation contains a small parameter in the highest derivative and a Fredholm operator perturbed by an operator addition on the right-hand side. Systems with small parameter in the highest derivative describe the motion of a viscous flow, the behavior of thin and flexible plates and shells, the process of a supersonic viscous gas flow around a blunt body, etc. The presence of a boundary layer phenomenon is revealed; in this case, even a small additive has a strong influence on the behavior of the solution. Asymptotic expansion of the solution in powers of small parameter is constructed by means of the Vasil'yeva-Vishik-Lyusternik method. Asymptotic property of the expansion is proved. To construct the regular part of the expansion, the equation decomposition method is used. It is consisted in a step-by-step transition to similar problems of decreasing dimensions.

Keywords: Cauchy problem, first-order differential equation, small parameter, Fredholm operator, boundary layer phenomenon, asymptotic expansion of solution, decomposition.

UDC: 517.928

Received: 21.01.2020

DOI: 10.20310/2686-9667-2020-25-129-48-56

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