Asymptotic Solution of the Cauchy Problem for a First-Order Equation with a Small Parameter in a Banach Space. The Regular Case

2018 ◽  
Vol 103 (3-4) ◽  
pp. 395-404 ◽  
Author(s):  
S. P. Zubova ◽  
V. I. Uskov
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Asymptotic Solution ◽  
Small Parameter ◽  
Order Equation ◽  
Regular Case ◽  
First Order ◽  
The Cauchy Problem

ASYMPTOTIC SOLUTION OF FIRST-ORDER EQUATION WITH SMALL PARAMETER UNDER THE DERIVATIVE WITH PERTURBED OPERATOR

2018 ◽  
pp. 784-796
Author(s):  
Vladimir Igorevich Uskov
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Cauchy Problem ◽  
Asymptotic Expansion ◽  
Asymptotic Solution ◽  
Small Parameter ◽  
Decomposition Method ◽  
Order Equation ◽  
First Order ◽  
The Cauchy Problem

The paper is devoted to the Cauchy problem for a differential equation with a small parameter when using a Fredholm operator in a Banach space with a certain method. The investigated effect of this parameter. The solution is in the form of an asymptotic expansion. When solving the problems of using the cascade decomposition method for equations, which allows us to split the equation into equations in subspaces.

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

2020 ◽  
pp. 48-56
Author(s):  
Vladimir I. Uskov
Keyword(s):  
Cauchy Problem ◽  
Small Parameter ◽  
Fredholm Operator ◽  
Gas Flow ◽  
First Order ◽  
Flexible Plates ◽  
The Cauchy Problem ◽  
Plates And Shells ◽  
Step Transition ◽  
The Right

We consider the Cauchy problem for a first-order differentialequation 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.

scholarly journals ON COMPLETENESS OF ROOT VECTORS OF THE CAUCHY PROBLEM OF THE FIRST ORDER EQUATION WITH DEVIATING ARGUMENT

2020 ◽  
Vol 2 (330) ◽  
pp. 50-57
Author(s):  
T.Sh. Kalmenov ◽  
◽  
A.Sh. Shaldanbayev ◽  
M.I. Akylbayev ◽  
A.N. Urmatova ◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Order Equation ◽  
Deviating Argument ◽  
First Order ◽  
The Cauchy Problem

Old and New Results for First Order Periodic ODEs without Uniqueness: a Comprehensive Study by Lower and Upper Solutions

2004 ◽  
Vol 4 (3) ◽  
Author(s):  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Lower And Upper Solutions ◽  
Elementary Approach ◽  
Qualitative Properties ◽  
First Order ◽  
The Past ◽  
The Cauchy Problem ◽  
Qualitative Properties Of Solutions ◽  
Comprehensive Study

AbstractAn elementary approach, based on a systematic use of lower and upper solutions, is employed to detect the qualitative properties of solutions of first order scalar periodic ordinary differential equations. This study is carried out in the Carathéodory setting, avoiding any uniqueness assumption, in the future or in the past, for the Cauchy problem. Various classical and recent results are recovered and generalized.

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