scholarly journals Nonlinear Singularly Perturbed Integro-Differential Equations and Regularization Method

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Regularization Method ◽  
Essential Role ◽  
Solution Space ◽  
Nonlinear Problems ◽  
Zero Point ◽  
Singularly Perturbed ◽  
Special Role ◽  
Limit Operator ◽  
Integrodifferential System ◽  
Singularly Perturbed Equations

The paper considers a nonlinear integro-differential system of singularly perturbed equations. We discuss the question of the spectrum of its operator, which does not coincide with the spectrum of its limit operator and includes an additionally identically zero point. In the case of linear systems, this difference does not play a special role, since the regularization and construction of the space of solutions of the corresponding iterative problems are realized at nonzero points of the spectrum. In the case of nonlinear problems, the identically zero point of the spectrum plays an essential role in the construction of the solution space in the resonance and nonresonance cases (see below); therefore, in most works using the regularization method in nonlinear problems, only the nonresonance case is usually considered. In the paper, for the classical integrodifferential system, regularization (according to Lomov) is carried out and the corresponding algorithm for constructing asymptotic solutions taking into account the zero point of the spectrum is developed.

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.

scholarly journals On ill-posedness concepts, stable solvability and saturation

2018 ◽  
Vol 26 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Bernd Hofmann ◽  
Robert Plato
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Solution Space ◽  
Nonlinear Problems ◽  
Special Role ◽  
Well Posedness ◽  
Operator Equations ◽  
Nonlinear Operators ◽  
Nonlinear Operator Equations ◽  
Linear Operator Equations

AbstractWe consider different concepts of well-posedness and ill-posedness and their relations for solving nonlinear and linear operator equations in Hilbert spaces. First, the concepts of Hadamard and Nashed are recalled which are appropriate for linear operator equations. For nonlinear operator equations, stable respective unstable solvability is considered, and the properties of local well-posedness and ill-posedness are investigated. Those two concepts consider stability in image space and solution space, respectively, and both seem to be appropriate concepts for nonlinear operators which are not onto and/or not, locally or globally, injective. Several example situations for nonlinear problems are considered, including the prominent autoconvolution problems and other quadratic equations in Hilbert spaces. It turns out that for linear operator equations, well-posedness and ill-posedness are global properties valid for all possible solutions, respectively. The special role of the nullspace is pointed out in this case. Finally, non-injectivity also causes differences in the saturation behavior of Tikhonov and Lavrentiev regularization of linear ill-posed equations. This is examined at the end of this study.

scholarly journals Regularized Asymptotics of the Solution of the Singularly Perturbed First Boundary Value Problem on the Semiaxis for a Parabolic Equation with a Rational “Simple” Turning Point

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 405
Author(s):  
Alexander Yeliseev ◽  
Tatiana Ratnikova ◽  
Daria Shaposhnikova
Keyword(s):  
Parabolic Equation ◽  
Maximum Principle ◽  
Boundary Value Problem ◽  
Regularization Method ◽  
Turning Point ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Asymptotic Convergence ◽  
The Maximum Principle ◽  
Regularized Asymptotics

The aim of this study is to develop a regularization method for boundary value problems for a parabolic equation. A singularly perturbed boundary value problem on the semiaxis is considered in the case of a “simple” rational turning point. To prove the asymptotic convergence of the series, the maximum principle is used.

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