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.
Received: 21.01.2020
Bibliographic databases:
Document Type:
Article
UDC:517.928
Language: Russian
Citation:
V. I. Uskov, “Asymptotic solution of the Cauchy problem for the first-order equation with perturbed Fredholm operator”, Russian Universities Reports. Mathematics, 25:129 (2020), 48–56
\Bibitem{Usk20}
\by V.~I.~Uskov
\paper Asymptotic solution of the Cauchy problem for the first-order equation with perturbed Fredholm operator
\jour Russian Universities Reports. Mathematics
\yr 2020
\vol 25
\issue 129
\pages 48--56
\mathnet{http://mi.mathnet.ru/vtamu169}
\crossref{https://doi.org/10.20310/2686-9667-2020-25-129-48-56}
\elib{https://elibrary.ru/item.asp?id=42655353}
Linking options:
https://www.mathnet.ru/eng/vtamu169
https://www.mathnet.ru/eng/vtamu/v25/i129/p48
This publication is cited in the following 2 articles:
V. I. Uskov, “Asimptotika resheniya uravneniya pervogo poryadka s malym parametrom pri proizvodnoi s kvadratichnym vozmuscheniem v pravoi chasti”, Vestnik TvGU. Seriya: Prikladnaya matematika, 2022, no. 1, 18–32
V. I. Uskov, “Asymptotic Solution of the Cauchy Problem for a First-Order Differential Equation with a Small Parameter in a Banach Space”, Math. Notes, 110:1 (2021), 145–151
Citation in format AMSBIB
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