On the regularized asymptotics of a solution to the Cauchy problem in the presence of a weak turning point of the limit operator

2021 ◽  
Vol 212 (10) ◽  
Author(s):  
Aleksandr Georgievich Eliseev
Keyword(s):  
Cauchy Problem ◽  
Turning Point ◽  
Limit Operator ◽  
The Cauchy Problem ◽  
Regularized Asymptotics

scholarly journals On the Regularized Asymptotics of a Solution to the Cauchy Problem in the Presence of a Weak Turning Point of the Limit Operator

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 86 ◽  
Author(s):  
Alexander Yeliseev
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Solution ◽  
Turning Point ◽  
Asymptotic Convergence ◽  
Limit Operator ◽  
The Cauchy Problem ◽  
Regularized Asymptotics

An asymptotic solution of the linear Cauchy problem in the presence of a “weak” turning point for the limit operator is constructed by the method of S. A. Lomov regularization. The main singularities of this problem are written out explicitly. Estimates are given for ε that characterize the behavior of singularities for ϵ→0. The asymptotic convergence of a regularized series is proven. The results are illustrated by an example. Bibliography: six titles.

scholarly journals Regularized Solution of Singularly Perturbed Cauchy Problem in the Presence of Rational “Simple” Turning Point in Two-Dimensional Case

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 124 ◽  
Author(s):  
Alexander Eliseev ◽  
Tatjana Ratnikova
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Solution ◽  
Regularization Method ◽  
Turning Point ◽  
Singularly Perturbed ◽  
Dimensional Case ◽  
Singularly Perturbed Problem ◽  
Two Dimensional ◽  
Limit Operator ◽  
Regularized Solution

By Lomov’s S.A. regularization method, we constructed an asymptotic solution of the singularly perturbed Cauchy problem in a two-dimensional case in the case of violation of stability conditions of the limit-operator spectrum. In particular, the problem with a ”simple” turning point was considered, i.e., one eigenvalue vanishes for t = 0 and has the form t m / n a ( t ) (limit operator is discretely irreversible). The regularization method allows us to construct an asymptotic solution that is uniform over the entire segment [ 0 , T ] , and under additional conditions on the parameters of the singularly perturbed problem and its right-hand side, the exact solution.

scholarly journals Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 138
Author(s):  
Tatiana Ratnikova
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Asymptotic Solution ◽  
Regularization Method ◽  
Turning Point ◽  
Singularly Perturbed ◽  
Stability Conditions ◽  
Asymptotic Convergence ◽  
Limit Operator ◽  
Operator Spectrum

The aim of the research is to develop the regularization method. By Lomov’s regularization method, we constructed a uniform asymptotic solution of the singularly perturbed Cauchy problem for a parabolic equation in the case of violation of stability conditions of the limit-operator spectrum. The problem with a “simple” turning point is considered in the case, when the eigenvalue vanishes at t=0 and has the form tm/na(t). The asymptotic convergence of the regularized series is proved.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1287-1293 ◽  
Author(s):  
Zujin Zhang ◽  
Dingxing Zhong ◽  
Shujing Gao ◽  
Shulin Qiu
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
The Cauchy Problem ◽  
Appl Math

In this paper, we consider the Cauchy problem for the 3D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves [S. Rahman, Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure, J. Comput. Appl. Math., 270 (2014), 88-99].

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

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