On the holomorphic regularization of singularly perturbed systems of differential equations

Kachalov, V. I.

doi:10.1134/s0965542517040054

Cited by 8 publications

(3 citation statements)

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“…. , z m with the variable z and apply the method of evaluating contour integrals described in [11]. As a result, (m + k) independent integrals of system (1) are constructed, being analytical at the point ε = 0:…”

Section: Construction Of the First Integrals For Tikhonov Systemsmentioning

confidence: 99%

Analytical Aspects of the Theory of Tikhonov Systems

Besova

Kachalov

2021

Mathematics

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Section: Construction Of the First Integrals For Tikhonov Systemsmentioning

confidence: 99%

Analytical Aspects of the Theory of Tikhonov Systems

Besova

Kachalov

2021

Mathematics

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“…The second part of our paper is devoted to the construction of approximate solutions of singularly perturbed problems using the method of holomorphic regularization [8,9]. The analysis of asymptotic methods for solving singularly perturbed problems shows that the solutions of such problems depend in two ways on a small parameter: regularly and singularly.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic and Pseudoholomorphic Solutions of Singularly Perturbed Differential and Integral Equations in the Lomov’s Regularization Method

Bobodzhanov

Safonov

Kachalov

2019

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We consider a singularly perturbed integral equation with weakly and rapidly varying kernels. The work is a continuation of the studies carried out previously, but these were focused solely on rapidly changing kernels. A generalization for the case of two kernels, one of which is weakly, and the other rapidly varying, has not previously been carried out. The aim of this study is to investigate the effects introduced into the asymptotics of the solution of the problem by a weakly varying integral kernel. In the second part of the work, the problem of constructing exact (more precise, pseudo-analytic) solutions of singularly perturbed problems is considered on the basis of the method of holomorphic regularization developed by one of the authors of this paper. The power series obtained with the help of this method for the solutions of singularly perturbed problems (in contrast to the asymptotic series constructed in the first part of this paper) converge in the usual sense.

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“…Следует также отметить, что обычная сходимость имеет место не только в регулярной теории возмущений Kachalov (в основном, благодаря теоремам Пуанкаре о разложении), но и в теории сингулярных возмущений (см. [1,2]). В работе будут рассмотрены начальная задача для уравнения Бюргерса, которое является одномерным аналогом уравнения Навье-Стокса, и уравнение Клейна-Гордона, возникающего в теории поля и дифференциальной геометрии (уравнение Sine-Гордона).…”

Section: Introductionunclassified

On the method of a small parameter in nonlinear mathematical physics

Kachalov¹,

Fedorov²

2018

Sib. Elektron. Mat. Izv.

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The method of a small parameter has been used in mathematical physics for a long time. However, with its help, in general, asymptotic solutions of differential equations are obtained. In the framework of the regularization method, S.A. Lomov proved that under certain restrictions on the data of the problem, one can obtain solutions in the form of series converging in the usual sense in powers of the small parameter, that is, solutions analytically dependent on the parameter. Here we consider two equations -the Burgers equation and the Klein-Gordon equation. The first of them represents a one-dimensional model of hydrodynamics, and the second one is considered in quantum field theory.

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On the holomorphic regularization of singularly perturbed systems of differential equations

Cited by 8 publications

References 1 publication

Analytical Aspects of the Theory of Tikhonov Systems

Analytical Aspects of the Theory of Tikhonov Systems

Asymptotic and Pseudoholomorphic Solutions of Singularly Perturbed Differential and Integral Equations in the Lomov’s Regularization Method

On the method of a small parameter in nonlinear mathematical physics

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