Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity

Bibulova, Dana; Kalimbetov, Burkhan; Safonov, V. F.

doi:10.3390/axioms11030141

Cited by 3 publications

(1 citation statement)

References 12 publications

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“…Singularly perturbed differential, integral and integro-differential equations with rapidly oscillating coefficients is considered in [24,25,26,27,28,29,30]. In the papers [31,32,33,34,35,36,37], regularized asymptotic solutions for linear singularly perturbed equations with rapidly changing kernels and rapidly oscillating inhomogeneities are studied and constructed. And in the work [38], an integral equation with a rapidly oscillating inhomogeneity is considered.…”

Section: Introductionmentioning

confidence: 99%

Influence of rapidly oscillating inhomogeneities in the formation of additional boundary layers for singularly perturbed integro-differential systems

2023

Advances in the Theory of Nonlinear Analysis and Its Application

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In this paper, the Lomov's regularization method is generalized to a singularly perturbed integro-differential equation with a fractional derivative and with a rapidly oscillating inhomogeneity. The main goal of the work is to reveal the influence of a rapidly changing kernel on the structure of the asymptotic of the problem solution and to study additional boundary functions that are generated by rapidly oscillating inhomogeneities.

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Section: Introductionmentioning

confidence: 99%

Influence of rapidly oscillating inhomogeneities in the formation of additional boundary layers for singularly perturbed integro-differential systems

2023

Advances in the Theory of Nonlinear Analysis and Its Application

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Influence of rapidly oscillating inhomogeneities in the formation of additional boundary layers for singularly perturbed integro-differential systems

Kalimbetov

2023

Advances in the Theory of Nonlinear Analysis and Its Application

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In this paper, the Lomov's regularization method is generalized to a singularly perturbed integro-differential equation with a fractional derivative and with a rapidly oscillating inhomogeneity. The main goal of the work is to reveal the influence of a rapidly changing kernel on the structure of the asymptotics of the problem solution and to study additional boundary functions that are generated by rapidly oscillating inhomogeneities.

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Asymptotic Solution of a Singularly Perturbed Integro-Differential Equation with Exponential Inhomogeneity

Kalimbetov

Safonov

Zhaidakbayeva

2023

Axioms

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The integro-differential Cauchy problem with exponential inhomogeneity and with a spectral value that turns zero at an isolated point of the segment of the independent variable is considered. The problem belongs to the class of singularly perturbed equations with an unstable spectrum and has not been considered before in the presence of an integral operator. A particular difficulty is its investigation in the neighborhood of the zero spectral value of inhomogeneity. Here, it is not possible to apply the well-known procedure of Lomov’s regularization method, so the authors have chosen the method of constructing the asymptotic solution of the initial problem based on the use of the regularized asymptotic solution of the fundamental solution of the corresponding homogeneous equation whose construction from the positions of the regularization method has not been considered so far. In the case of an unstable spectrum, it is necessary to take into account its point features. In this case, inhomogeneity plays an essential role. It significantly affects the type of singularities in the solution of the initial problem. The fundamental solution allows us to construct asymptotics regardless of the nature of the inhomogeneity (it can be both slowly changing and rapidly changing, for example, rapidly oscillating). The approach developed in the paper is universal with respect to arbitrary inhomogeneity. The first part of the study develops an algorithm for the regularization method to construct the asymptotic (of any order on the parameter) of the fundamental solution of the corresponding homogeneous integro-differential equation. The second part is devoted to constructing the asymptotics of the solution of the original problem. The main asymptotic term is constructed in detail, and the possibility of constructing its higher terms is pointed out. In the case of a stable spectrum, we can construct regularized asymptotics without using a fundamental solution.

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Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity

Cited by 3 publications

References 12 publications

Influence of rapidly oscillating inhomogeneities in the formation of additional boundary layers for singularly perturbed integro-differential systems

Influence of rapidly oscillating inhomogeneities in the formation of additional boundary layers for singularly perturbed integro-differential systems

Influence of rapidly oscillating inhomogeneities in the formation of additional boundary layers for singularly perturbed integro-differential systems

Asymptotic Solution of a Singularly Perturbed Integro-Differential Equation with Exponential Inhomogeneity

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