scholarly journals On the Right Weak Solution of the Cauchy Problem for a Quasilinear Equation of First Order

1969 ◽  
Vol 19 (6) ◽  
pp. 483-487 ◽  
Author(s):  
Eberhard Hopf
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Quasilinear Equation ◽  
First Order ◽  
The Cauchy Problem ◽  
The Right

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.

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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