A Numerical Method for Solving a System of Hypersingular Integral Equations of the Second Kind

Kostenko, A. V.

doi:10.1007/s10559-016-9840-3

Cited by 5 publications

(3 citation statements)

References 4 publications

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“…The presented results continue, develop, and are based on the series of works [11,12,16,18] devoted to the qualitative theory of hypersingular integral equations and numerical methods. Another standing shoulder for the presented results is the book [22].…”

Section: Introductionmentioning

confidence: 66%

A Numerical Method for Solving a Complete Hypersingular Integral Equation of the Second Kind and Its Justification

Kostenko

2023

Mathematical Modelling and Analysis

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A complete hypersingular integral equation of the second kind was obtained as a boundary integral equation for the diffraction and scattering problem of electromagnetic waves in space separated by the periodically placed non-perfectly conducting strips. The equation includes a singular integral that distinguishes it from the studied second-kind hypersingular equation. Our motivation is the need to have a numerical method for the equation, its applicability borders, and guaranteed convergence. The numerical method has the type of Nyström. The justification of the method envelops a proof of the theorem of existence and uniqueness of the solution and an estimate of the convergence rate of sequence of the approximate solutions to an exact solution.

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Section: Introductionmentioning

confidence: 66%

A Numerical Method for Solving a Complete Hypersingular Integral Equation of the Second Kind and Its Justification

Kostenko

2023

Mathematical Modelling and Analysis

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“…Завдяки універсальності методу параметричного подання інтегральних перетворень у періодичному випадку були отримані три різні моделі для опису цього випадку: дві на базі сингулярних інтегральних рівнянь різного типу та одна гіперсинулярна модель [15][16][17]. Математичне обґрунтовування чисельного розв'язання таких рівнянь було дано у роботах [18][19][20]. Отриманні за допомогою різних підходів результати добре узгоджені між собою.…”

Section: вступunclassified

Mathematical Model of Wave Diffraction by the System of Stripes With Different Values of Surface Impedance

Dushkin

Мельник

2021

JNAM

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A mathematical model of diffraction of E-polarized and H-polarized waves on a finite system of not perfectly conducting tapes is obtained. The value of the surface impedance on the two sides of the stripes is different. The initial boundary value problem for the Helmholtz equation with boundary conditions of the third kind was reduced to a system of boundary integral equations. This system of boundary integral equations consists of singular integral equations of the first kind and integral equations of the second kind with a logarithmic singularity. The method of parametric representation of integral operator was used to perform transformations. The values of the physical characteristics of the process are expressed through the solutions of the obtained systems of integral equations. Numerical solution of these equations is performed using a computational scheme based on the discrete singularities method.

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“…The method of parametric representations of integral operators makes it possible to obtain systems of integral equations of other types [29], [30]. In particular, the original boundary-value problem was reduced to a system consisting of hypersingular integral equations of the second kind and the Fredholm integral equation of the second kind [32], [37]. Therefore, it is a good base of interest for a comparative computer experiment.…”

Section: Origins Of Researchmentioning

confidence: 99%

Discrete mathematical model of the scattering process of E-polarized wave on a periodic impedance grating

2019

MAMM

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The method of numerical modeling of wave scattering by periodic impedance grating is considered. In the case of a harmonic dependence of the field on time and the uniformity of the structure along a certain axis, the three-dimensional problem reduces to considering of two 2D problems for the components of the E-polarized and H-polarized waves. The signle nonzero component of the electric field created by the incident E-polarized wave is the solution of the boundary value problem for the Helmholtz equation with Robin boundary conditions. It follows from the physical formulation of the problem that its solutions satisfy the Floquet quasiperiodicity condition, the condition of finiteness of energy in any bounded region of the plane. Also, the difference between the total and incident fields satisfies the Sommerfeld radiation condition. Following the ideas of the works of Yu.V. Gandel, using the method of parametric representations of integral operators, the boundary-value problem reduces to two systems of integral equations. The first one is the system of singular equations of the first kind with additional integral conditions. The second system consists of the Fredholm boundary integral equations of the second kind with a logarithmic singularity in the integrand. A discrete model for various values of the discretization parameter is equivalent to systems of singular integral equations. By solving these equations, approximate values of the main field characteristics are determined. The method of parametric representations of integral operators makes it possible to obtain systems of integral equations of other types. In particular, the initial boundary-value problem reduces to a system consisting of hypersingular integral equations of the second kind and the Fredholm integral equation of the second kind. A numerical experiment was conducted for cases of different location of tapes. Calculations were performed for the proposed model and the model based on hypersingular equations. They showed the closeness of the obtained results in a wide range of parameters studied.

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A Numerical Method for Solving a System of Hypersingular Integral Equations of the Second Kind

Cited by 5 publications

References 4 publications

A Numerical Method for Solving a Complete Hypersingular Integral Equation of the Second Kind and Its Justification

A Numerical Method for Solving a Complete Hypersingular Integral Equation of the Second Kind and Its Justification

Mathematical Model of Wave Diffraction by the System of Stripes With Different Values of Surface Impedance

Discrete mathematical model of the scattering process of E-polarized wave on a periodic impedance grating

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