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

Yeliseev, Alexander

doi:10.3390/axioms9030086

Cited by 6 publications

(2 citation statements)

References 5 publications

(10 reference statements)

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“…Moreover, there is still no complete mathematical theory for singularly perturbed problems with an unstable spectrum, although they began to be studied from a general mathematical standpoint about 50 years ago. Of particular interest among such problems are those in which the spectral features are expressed in the form of point instability (see, for example, [9][10][11][12]). In works devoted to singularly perturbed problems, some of the features of this type are called turning points, and their classification is as follows:…”

Section: Introductionmentioning

confidence: 99%

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Regularized Asymptotics of the Solution of a Singularly Perturbed Mixed Problem on the Semiaxis for the Schrödinger Equation with the Potential Q = X2

Yeliseev,

Ratnikova,

Shaposhnikova

2023

Mathematics

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In this paper, we study the solution of a singularly perturbed inhomogeneous mixed problem on the half-axis for the Schrödinger equation in the presence of a “strong” turning point for the limit operator on time interval that do not contain focal points. Based on the ideas of the regularization method for asymptotic integration of problems with an unstable spectrum, it is shown how regularizing functions should be constructed for this type of singularity. The paper describes in detail the formalism of the regularization method, justifies the algorithm, constructs an asymptotic solution of any order in a small parameter, and proves a theorem on the asymptotic convergence of the resulting series.

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Section: Introductionmentioning

confidence: 99%

“…Here, we give links to several recent studies in the framework of the method of regularization of singularly perturbed problems with singularities in the spectrum of the limit operator of the indicated form: for a simple turning point, see papers [9,10], for a weak turning point, see [11], and for a strong turning point, see [12,13].…”

Section: Introductionmentioning

confidence: 99%

Regularized Asymptotics of the Solution of a Singularly Perturbed Mixed Problem on the Semiaxis for the Schrödinger Equation with the Potential Q = X2

Yeliseev,

Ratnikova,

Shaposhnikova

2023

Mathematics

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Regularized asymptotics of the solution of systems of parabolic differential equations

Omuraliev

Abylaeva

2022

Filomat

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The regularization method for singularly perturbed problems of S. A. Lomov is generalized to constructing the asymptotics of the solution of the first boundary value problem for systems of differential equations of parabolic type with a small parameter at all derivatives.It is shown that the asymptotics of the solution of the problem contains n exponential, 2n parabolic and 2n angle boundary layer functions. The exponential boundary layer function describes the boundary layer along t = 0, the boundary layer along x = 0 and x = 1 is described by parabolic boundary layer functions.

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

Bibulova

Kalimbetov

Safonov

2022

Axioms

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This article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form of a Volterra-type integral. The presence of an integral operator and its type significantly affect the development of an algorithm for asymptotic solutions, in the implementation of which it is necessary to take into account essential singularities generated by the rapidly decreasing kernel of the integral operator. It is shown in tise work that when passing the structure of essentially singular singularities changes from an integral operator of Volterra type to an operator of Fredholm type. If in the case of the Volterra operator they change with a change in the independent variable, then the singularities generated by the kernel of the integral Fredholm-type operators are constant and depend only on a small parameter. All these effects, as well as the effects introduced by the rapidly oscillating inhomogeneity, are necessary to take into account when developing an algorithm for constructing asymptotic solutions to the original problem, which is implemented in this work.

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

Cited by 6 publications

References 5 publications

Regularized Asymptotics of the Solution of a Singularly Perturbed Mixed Problem on the Semiaxis for the Schrödinger Equation with the Potential Q = X2

Regularized Asymptotics of the Solution of a Singularly Perturbed Mixed Problem on the Semiaxis for the Schrödinger Equation with the Potential Q = X2

Regularized asymptotics of the solution of systems of parabolic differential equations

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

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