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 singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity

2021 ◽  
Vol 19 (1) ◽  
pp. 244-258
Author(s):  
Burkhan T. Kalimbetov ◽  
Olim D. Tuychiev
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Asymptotic Solution ◽  
Regularization Method ◽  
Singularly Perturbed ◽  
New Approach ◽  
The Cauchy Problem ◽  
Oscillating Coefficients ◽  
Oscillating Coefficient ◽  
The Relationship

Abstract In this paper, the regularization method of S. A. Lomov is generalized to integro-differential equations with rapidly oscillating coefficients and with a rapidly oscillating right-hand side. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution of this problem. The case of coincidence of the frequencies of a rapidly oscillating coefficient and a rapidly oscillating inhomogeneity is considered. In this case, only the identical resonance is observed in the problem. Other cases of the relationship between frequencies can lead to so-called non-identical resonances, the study of which is nontrivial and requires the development of a new approach. It is supposed to study these cases in our further work.

scholarly journals Differential Equation in a Banach Space Multiplicatively Perturbed by Random Noise

2017 ◽  
Vol 63 (4) ◽  
pp. 599-614
Author(s):  
V G Zadorozhniy ◽  
M A Konovalova
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Random Noise ◽  
Random Coefficients ◽  
First Order ◽  
The Cauchy Problem ◽  
Moment Functions ◽  
Nonhomogeneous Differential Equation ◽  
Variational Derivatives ◽  
The Moment

We consider the problem of finding the moment functions of the solution of the Cauchy problem for a first-order linear nonhomogeneous differential equation with random coefficients in a Banach space. The problem is reduced to the initial problem for a nonrandom differential equation with ordinary and variational derivatives. We obtain explicit formula for the mathematical expectation and the second-order mixed moment functions for the solution of the equation.

scholarly journals Optimal time and space regularity for solutions of degenerate differential equations

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Alberto Favaron
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Optimal Time ◽  
Time And Space ◽  
Optimal Regularity ◽  
Degenerate Differential Equation ◽  
The Cauchy Problem ◽  
Degenerate Differential Equations

AbstractWe derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.

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