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Boundary integral equations of diffraction problems on flat screens and their discrete mathematical models


Y.V. Gandel, V.S. Bylugin

A problem of electromagnetic wave diffraction on a perfectly conducting plane screen that is situated on a boundary between two media with different permittivity is investigated. Electromagnetic wave scattering by perfectly conducting screen is situated in a heterogeneous space is one of the problems studied in sufficient detail . That is not the a case, when the screen is situated in a heterogeneous space, although in practice it is often necessary to consider the situation, when the screen is located on the boundary between two media. It is necessary to answer similar questions in many cases, such as synthesis of antenna systems and electrical exploration of mineral resources. There are two unknowns function and , using which fields и in far-field and near-field regions are expressed. The equivalence between corresponding Maxwell equations and pseudodifferential equation system for unknown function Fourier transform is proved. In this system the pseudodifferential operators with increasing and decreasing symbols are derived. It is proved, that the operators with increasing symbols are hypersingular integral operators and operators with decreasing symbols are integral operators with smooth kernels. Thus, using parametric presentation of pseudodifferential operators, the system reduces to a system of two hypersingular integral equations for the unknowns and . Describing discrete mathematical model of the problem it should be noted that the coordinate plane, in which the screen is lo-cated, is represented as a union of equal squares with a side length of . The set of squares , which belong to a region , are denoted . The approximation of unknown function и are sought as piecewise constant functions defined on that equal and accordingly on each square . Finally, we approximate solution search reduced to a linear algebraic equation systems for and . The numerical experiment results are given for Sierpinski's carpet different orders.

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