On solvability of homogeneous Riemann–Hilbert problem with discontinuities of coefficients and two-sided curling at infinity of a logarithmic order

Салимов, Р. Б.; Шабалин, П. Л.

doi:10.3103/s1066369x16010047

Cited by 4 publications

(4 citation statements)

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“…Methods of the theory of analytical functions are widely used in practice [22,[26][27][28][29]. Due to the importance of boundary value problems, research is conducted in the [30,31], and also solutions in the case of special behavior of the boundary value coefficients [32]. In these works the behavior of singular integrals at special points [33] and peculiarities of application of the Christoffel-Schwarz formula [34] are used.…”

Section: Resultsmentioning

confidence: 99%

Explosive technologies in transport construction

Tuktamyshov

2021

E3S Web Conf.

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Performing different works (construction of road embankments, canals, tunnels, moving some soil mass in a given direction), as well as construction of various towers, power transmission supports, foundations for bridge supports, special pits for anchors are carried out by explosive technologies. The paper analyzes in detail a new method of determining the boundary of the explosion excavation. Its distinguishing features are the use of an impulse-hydrodynamic model for setting explosion problems and use of boundary problems of the theory of analytical functions. The new method is used in solving the problem of determining the boundary of the ejection excavation for the case of a buried charge. The results of calculations for the main geometric parameters of the explosion are given. The prospects for the development and application of the method are discussed.

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Section: Resultsmentioning

confidence: 99%

Explosive technologies in transport construction

Tuktamyshov

2021

E3S Web Conf.

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“…Отметим, что регулярные составляющие в правых частях формул (11), (12) являются голоморфными в соответствующих областях функциями. Итак, мы доказали, что при условиях (3), (4) функция 𝐼 ε (𝑧) равномерно сходится на компакте к пределу (10).…”

Section: решение эллиптической системыunclassified

“…Обозначим односторонние пределы функции 𝑎 0 (𝑥) в начале координат и в бесконечности символами: 𝑎 0 (0+) = 𝑖µ + , 𝑎 0 (0−) = 𝑖µ − , 𝑎 0 (+∞) = 𝑖ν + , 𝑎 0 (−∞) = 𝑖ν − . В соответствии с равенствами ( 14), (11), (12) для функций Ω(𝑧) вблизи начала координат справедлива формула…”

Section: решение эллиптической системыunclassified

“…Пусть для коэффициентов уравнения (1) выполнены условия ( 2)-( 5) и (𝑒 −Ω 𝐹 )(𝑧) ∈ 𝐿 𝑝,2 (𝐸), 𝑝 > 2, односторонние пределы на бесконечности функции 𝑎 0 (𝑥) удовлетворяют системе неравенств (26). Тогда, если хотя бы одно из неравенств (26) строгое, а правая часть краевого условия (17) имеет асимптотику (20), то задача Гильберта (17) имеет бесконечное множество решений 𝑈 (𝑧), определенных формулой (12) с ϕ(𝑧) в виде (35). Если же в (26) ν − − ν + = 0, µ − − µ + = 0, то задача имеет единственное с точностью до постоянного вещественного сомножителя решение и определяется формулой (12) с ϕ(𝑧) в виде (35), в которой 𝐹 (𝑧) = const.…”

Section: неоднородная задачаunclassified

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Hilbert’s Boundary Value Problem for a Class of Generalized Analytic Functions with a Singular Line

Svetlov¹

2022

MPCS

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In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on the real axis for one generalized equation Cauchy — Riemann with singular coefficient. A structural formula for the general solution of this equation is obtained under constraints leading to an infinite index of the accompanying Hilbert boundary value problem for analytic functions. The study of the solvability of the latter is the basis for solving the boundary value problem for generalized analytical functions. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in terms of the characteristics of the coefficient of the boundary condition. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in the class of bounded analytic functions. This study made it possible to obtain a formula for solving the Hilbert problem already for generalized analytic functions. We also formulated the conditions under which the problem has no solution, has a single solution or an infinite number of solutions.

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The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

Shabalin,

Faizov

2024

jour

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This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

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On solvability of homogeneous Riemann–Hilbert problem with discontinuities of coefficients and two-sided curling at infinity of a logarithmic order

Cited by 4 publications

References 5 publications

Explosive technologies in transport construction

Explosive technologies in transport construction

Hilbert’s Boundary Value Problem for a Class of Generalized Analytic Functions with a Singular Line

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

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