THE SCATTERING OF A PLANE WAVE BY A ROW OF SMALL CYLINDERS

1960 ◽  
Vol 38 (2) ◽  
pp. 272-289 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Boundary Condition ◽  
Plane Wave ◽  
Integral Equations ◽  
Cross Sections ◽  
Simultaneous Equations ◽  
Wave Number ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Higher Order Terms ◽  
Scattering Cross Sections

Consideration is given to the scattering of a plane wave by N cylinders equispaced in a row. The problems associated with scatterers, both "soft" and "hard" in the acoustical sense, are treated. An application of Green's theorem together with the appropriate boundary condition on the cylinders leads to a set of simultaneous integral equations in the unknown function on the cylinders.Solutions in the form of series in powers of a small parameter δ (essentially the ratio of cylinder dimension to wavelength) are assumed. In the case of elliptic cylinders, the integral equations are reduced to sets of linear algebraic equations. Only for the first term in the solution for "soft" cylinders is it necessary to solve N simultaneous equations in N unknowns; all other equations involve essentially only one unknown. Far-fields and scattering cross sections are calculated. The case of two "soft" cylinders is given particular attention.Conditions for justification of the neglect of higher-order terms are discussed. It is found that all terms but the first (in either problem) may be neglected if [Formula: see text] and (N–1)/(ka) is sufficiently small. (Here a is the spacing between centers of adjacent cylinders, and k is the wave number.) For this reason these solutions are most useful when the number of cylinders is small.

Electromagnetic plane wave diffraction by a cylindrical arc with edges: H-polarized case

2021 ◽  
pp. 1-15
Author(s):  
Kamil Karaçuha ◽  
Vasil Tabatadze ◽  
Eldar Ismailovich Veliyev
Keyword(s):  
Plane Wave ◽  
Cross Sections ◽  
Inversion Method ◽  
Algebraic Equations ◽  
Method Of Moment ◽  
Electromagnetic Plane Wave ◽  
Radar Cross Sections ◽  
Linear Algebraic Equations ◽  
Orthogonal Set ◽  
Electromagnetic Plane

An accurate hybrid method (numerical-analytical method) for the diffraction of H-polarized electromagnetic plane wave by perfectly electric conducting cylindrical bodies containing edges and a longitudinal slit aperture is proposed. This method is the combination of the Method of Moment and semi-inversion method. The current density function is expressed as the Chebyshev polynomials forming a complete orthogonal set of basis functions. Then, the initial problem is reduced to a system of linear algebraic equations. After inversion, the unknown coefficients are obtained. Then, near and far-field distributions, radar cross-sections are obtained. The resonances are observed for different values of the aperture size, radius of the arc, and the results are compared with previous outcomes.

scholarly journals On Solvability Conditions for a Certain Conjugation Problem

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 234
Author(s):  
Vladimir Vasilyev ◽  
Nikolai Eberlein
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Mellin Transform ◽  
Algebraic Equations ◽  
Conjugation Problem ◽  
Solvability Conditions ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

A Boundary Element Method for Plane Anisotropic Elastic Media

1990 ◽  
Vol 57 (3) ◽  
pp. 600-606 ◽  
Author(s):  
Kyu J. Lee ◽  
A. K. Mal
Keyword(s):  
Integral Equations ◽  
Complex Plane ◽  
Singular Integral Equations ◽  
Boundary Curve ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Numerical Formulation ◽  
Complex Linear ◽  
Anisotropic Elastic ◽  
Improved Accuracy

The general problem of plane anisotropic elastostatics is formulated in terms of a system of singular integral equations with Cauchy kernels by means of the classical stress function approach. The integral equations are represented over the image of the boundary in the complex plane and a numerical scheme is developed for their solution. The boundary curve is discretized and suitable polynomial approximations of the unknown functions in terms of the complex variable are introduced. This reduces the equations to a set of complex linear algebraic equations which can be inverted to yield the stresses in a straightforward manner. The major difference between the present technique and the previous ones is in the numerical formulation. The integral equations are discretized in the complex plane and not in terms of real variables which depend on arc length, resulting in improved accuracy in presence of strong boundary curvature.

Modification of scattering of waves by central loading

1968 ◽  
Vol 46 (24) ◽  
pp. 2755-2763 ◽  
Author(s):  
Chin-Lin Chen
Keyword(s):  
Plane Wave ◽  
Cross Section ◽  
Cross Sections ◽  
Normal Incidence ◽  
Physical Explanation ◽  
Lumped Element ◽  
Scattering Cross ◽  
Back Scattering ◽  
Conducting Wire ◽  
Scattering Cross Sections

The problem of the scattering of a plane wave by a long, thin, perfectly conducting wire is studied. The scatterer is loaded at its center by a lumped element. The effects of the loading on the scattering of waves are investigated. Numerical results are obtained for the case of normal incidence. The results show that for relatively short wires, the back-scattering cross sections may be modified effectively by central loading, while for longer wires, the modification is rather difficult to achieve. To nullify the back-scattering cross section completely, it is necessary to use active loading if kh > 3.6. A physical explanation is also presented.

scholarly journals A convergence theorem for singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 539-552 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Convergence Theory ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

Isogeometric analysis of boundary integral equations: High-order collocation methods for the singular and hyper-singular equations

2016 ◽  
Vol 26 (08) ◽  
pp. 1447-1480 ◽  
Author(s):  
Matthias Taus ◽  
Gregory J. Rodin ◽  
Thomas J. R. Hughes
Keyword(s):  
Integral Equations ◽  
Isogeometric Analysis ◽  
Computer Aided Design ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
High Order ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Integration Schemes ◽  
Linear Algebraic Equations

Isogeometric analysis is applied to boundary integral equations corresponding to boundary-value problems governed by Laplace’s equation. It is shown that the smoothness of geometric parametrizations central to computer-aided design can be exploited for regularizing integral operators to obtain high-order collocation methods involving superior approximation and numerical integration schemes. The regularization is applicable to both singular and hyper-singular integral equations, and as a result one can formulate the governing integral equations so that the corresponding linear algebraic equations are well-conditioned. It is demonstrated that the proposed approach allows one to compute accurate approximate solutions which optimally converge to the exact ones.

scholarly journals Numerical Solution for the 2D Linear Fredholm Functional Integral Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Neda Khaksari ◽  
Mahmoud Paripour ◽  
Nasrin Karamikabir
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Unknown Function ◽  
Numerical Solutions ◽  
Functional Integral ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

In this work, a numerical method is applied for obtaining numerical solutions of Fredholm two-dimensional functional linear integral equations based on the radial basis function (RBF). To find the approximate solutions of these types of equations, first, we approximate the unknown function as a finite series in terms of basic functions. Then, by using the proposed method, we give a formula for determining the unknown function. Using this formula, we obtain a numerical method for solving Fredholm two-dimensional functional linear integral equations. Using the proposed method, we get a system of linear algebraic equations which are solved by an iteration method. In the end, the accuracy and applicability of the proposed method are shown through some numerical applications.

scholarly journals Solving of Linear Volterra-Fredholm Integral Equations via Modification of Block Pulse Functions

2021 ◽  
Vol 17 (1) ◽  
pp. 33
Author(s):  
Ayyubi Ahmad
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Triangular System ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Linear Algebraic Equations

A computational method based on modification of block pulse functions is proposed for solving numerically the linear Volterra-Fredholm integral equations. We obtain integration operational matrix of modification of block pulse functions on interval [0,T). A modification of block pulse functions and their integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of  linear Volterra-Fredholm integral equations. Furthermore, the rate of convergence is  O(h) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.

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