A comparison of integral equations with unique solution in the resonance region for scattering by conducting bodies

Для численного решения классической задачи дифракции электромагнитной волны на идеально проводящих объектах используется метод граничных интегральных уравнений с гиперсингулярными интегралами, к которым применяются метод кусочно-постоянных аппроксимаций и метод коллокации. В результате задача сводится к системе линейных уравнений, коэффициециенты которой выражаются через интегралы по ячейкам разбиения с сильной степенной особенностью. Для вычисления этих интегралов применяется развитый ранее подход, основанный на выделении в явном виде членов с сильной особенностью, вычисляемых аналитически. В рамках этого подхода в настоящей статье протестирована численная схема, в которой вычисление оставшихся членов со слабосингулярными интегралами по ячейкам разбиения осуществляется путем построения более мелкой сетки второго уровня с домножением подынтегрального выражения на сглаживающий множитель. На примере задачи дифракции на теле в форме прямоугольного крыла показано, что такая схема, в частности, позволяет решать задачи дифракции на телах малой толщины. При этом толщина тела может быть даже меньше диаметра ячеек основного разбиения, но при условии, что диаметр ячеек сетки второго уровня существенно меньше, чем толщина тела. The method of boundary integral equations with hypersingular integrals is used for the numerical solution of the classical problem of electromagnetic wave scattering on ideally conducting bodies. The corresponding integral equations are solved by the methods of piecewise constant approximations and collocation. As a result, the problem is reduced to a system of linear algebraic equations whose coefficients are expressed in terms of integrals over partition cells with a strong power singularity. These integrals are evaluated using the previously developed approach based on the extraction of terms with a strong singularity calculated analytically. The proposed numerical scheme based on the calculation of the remaining terms with weakly singular integrals over partition cells is performed by constructing a fine grid of second level with multiplication of the integrands on a smoothing factor is tested. By the example of scattering on a rectangular it is shown, in particular, that this scheme allows one to solve the scattering problem on bodies of small thickness. In this case, the thickness of a body may be less then the diameter of the first level cells. However, the diameter of the second level cells must be much less than the thickness of the body.

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Solution of system of the Fredholm integral equation of the first kind

Modern Innovations, Systems and Technologies ◽

10.47813/2782-2818-2021-1-4-1-7 ◽

2021 ◽

Vol 1 (4) ◽

pp. 1-7

Author(s):

Vladimir Uskov

Keyword(s):

Integral Equation ◽

Unique Solution ◽

Integral Equations ◽

Fredholm Integral Equation ◽

Production Costs ◽

Fredholm Integral Equations ◽

System Of Equations ◽

Blurred Image ◽

Basic Functions ◽

Image Production

The article is devoted to the study of a system of two inhomogeneous Fredholm integral equations of the first kind with two required functions depending on one variable. Integral equations describe the restoration of a blurred image, production costs, etc. Fredholm integral equations with one desired function have been considered in many works, but relatively few works have been devoted to systems of such equations. The questions of stability for the solution of systems and the construction of a regularizing system of equations were investigated, but the solution was not constructed in an explicit form. In this paper, the kernels depend on two variables. The case is considered: in the kernels and inhomogeneities, the variables are separated in the equations; these functions are decomposed on the basis of two functions on the interval of integration. Examples of basic functions are given. A condition is determined under which the system has a unique solution in the chosen basis, formulated as a theorem. The solution is found in the form of an expansion in this basis. To illustrate the results obtained, an example is considered

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FIXED POINT RESULTS IN COMPLEX VALUED METRIC SPACES WITH AN APPLICATION

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi190313018r ◽

2021 ◽

pp. 237

Author(s):

Rashwan A. Rashwan ◽

Hasanen A. Hammad ◽

Liliana Guran

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Unique Solution ◽

Integral Equations ◽

Point Theorem ◽

Metric Spaces ◽

Contractive Condition ◽

Urysohn Integral Equations ◽

Complex Valued

In this paper, we introduce fixed point theorem for a general contractive condition in complex valued metric spaces. Also, some important corollaries under this contractive condition areobtained. As an application, we find a unique solution for Urysohn integral equations and some illustrative examples are given to support our obtaining results. Our results extend and generalize the results of Azam et al. [2] and some other known results in the literature.

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The Solutions of Mixed Monotone Fredholm-Type Integral Equations in Banach Spaces

Discrete Dynamics in Nature and Society ◽

10.1155/2013/604105 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Hua Su

Keyword(s):

Boundary Value Problem ◽

Banach Spaces ◽

Unique Solution ◽

Integral Equations ◽

Nonlinear Ordinary Differential Equations ◽

Point Boundary ◽

Fredholm Type ◽

Monotone Iterative ◽

Type Integral ◽

Ordered Banach Spaces

By introducing new definitions ofϕconvex and-φconcave quasioperator andv0quasilower andu0quasiupper, by means of the monotone iterative techniques without any compactness conditions, we obtain the iterative unique solution of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces. Our results are even new toϕconvex and-φconcave quasi operator, and then we apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations in the ordered Banach spaces.

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