scholarly journals MATHEMATICAL MODEL OF WAVE DIFFRACTION BY THE SYSTEM OF STRIPES WITH DIFFERENT VALUES OF SURFACE IMPEDANCE

2021 ◽  
pp. 131-137
Author(s):  
V. D. Dushkin ◽  
V. N. Melnik
Keyword(s):  
Mathematical Model ◽  
Integral Equations ◽  
Surface Impedance ◽  
Boundary Integral Equations ◽  
Logarithmic Singularity ◽  
Singular Integral Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Parametric Representation ◽  
Boundary Integral

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.

scholarly journals APPLICATION OF THE METHOD OF BOUNDARY INTEGRAL EQUATIONS FOR NON-STATIONARY PROBLEM OF THERMAL CONDUCTIVITY IN AXIALLY SYMMETRIC DOMAIN

2017 ◽  
Vol 1 ◽  
pp. 61-68
Author(s):  
Grigoriy Zrazhevsky ◽  
Vera Zrazhevska
Keyword(s):  
Thermal Conductivity ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Symmetric Domain ◽  
Singular Integral Equations ◽  
Initial Boundary ◽  
Boundary Integral ◽  
Fourier Series Expansion ◽  
Axially Symmetric ◽  
Initial Boundary Problem

The article considers the non-stationary initial-boundary problem of thermal conductivity in axially symmetric domain in Minkowski space, formulated as equivalent boundary integral equation. Using the representation of the solution in the form of a Fourier series expansion, the problem is reformulated as an infinite system of two-dimensional singular integral equations regarding expansion coefficients. The paper presents and investigates the explicit form for fundamental solutions used in the integral representation of the solution in the domain and on the border. The obtained results can be used in the construction of efficient numerical boundary element method for estimation of structures behavior under the influence of intense thermal loads in real-time.

Reflection and Transmission of a Plane Wave by an Array of Cavities in the Interface of Two Solids

1992 ◽  
Vol 59 (1) ◽  
pp. 102-108 ◽  
Author(s):  
Yonglin Xu
Keyword(s):  
Plane Wave ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Singular Integral Equations ◽  
Boundary Integral ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Typical Cell ◽  
Wave States ◽  
Doubly Periodic Array

The reflection and transmission of a plane wave by a distribution of cavities in the interface of two solids of different mechanical properties are investigated. For the calculation of the reflection and transmission coefficients by a distribution of cavities, six auxiliary wave states are used in conjunction with the reciprocal identity. Specific results are presented for scattering by a doubly periodic array of cavities in the interface of solids of different elastic moduli and mass densities. For a typical cell, the boundary integral equations for scattering by a cavity at the interface of two solids are derived on the basis of continuity of displacements and tractions across the interface and by taking advantage of the geometrical periodicity. Solutions to the system of singular integral equations have been obtained by the boundary element method. Numerical results are presented as functions of the frequency for two angles of incidence.

scholarly journals On a nonclassical problem for the heat equation and the Feller semigroup generated by it

2020 ◽  
Vol 12 (2) ◽  
pp. 297-310
Author(s):  
B.I. Kopytko ◽  
A.F. Novosyadlo
Keyword(s):  
Boundary Integral Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Continuous Functions ◽  
Boundary Integral ◽  
Initial Boundary Value Problems ◽  
Feller Semigroup ◽  
Initial Boundary Value ◽  
Bounded Continuous Functions

The initial boundary value problem for the equation of heat conductivity with the Wenzel conjugation condition is studied. It does not fit into the general theory of parabolic initial boundary value problems and belongs to the class of conditionally correct ones. In space of bounded continuous functions by the method of boundary integral equations its classical solvability under some conditions is established. In addition, it is proved that the obtained solution is a Feller semigroup, which represents some homogeneous generalized diffusion process in the area considered here.

Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle

1993 ◽  
Vol 123 (1) ◽  
pp. 185-208 ◽  
Author(s):  
P. A. Martin ◽  
Petri Ola
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Electromagnetic Waves ◽  
Boundary Integral Equations ◽  
Surrounding Medium ◽  
Singular Integral Equations ◽  
Boundary Integral ◽  
Tangential Vector ◽  
Single Integral

SynopsisTime-harmonic electromagnetic waves are scattered by a homogeneous dielectric obstacle. The corresponding electromagnetic transmission problem is reduced to a single integral equation over S for a single unknown tangential vector field, where S is the interface between the obstacle and the surrounding medium. In fact, several different integral equations are derived and analysed, including two previously-known equations due to E. Marx and J. R. Mautz, and two new singular integral equations. Mautz's equation is shown to be uniquely solvable at all frequencies. A new uniquely solvable singular integral equation is also found. The paper also includes a review of methods using pairs of coupled integral equations over S. It is these methods that are usually used in practice, although single integral equations seem to offer some computational advantages.

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

2019 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Logarithmic Singularity ◽  
Boundary Value ◽  
Integral Operators ◽  
Radiation Condition ◽  
Boundary Integral ◽  
Wide Range ◽  
Systems Of Integral Equations

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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