THE TWO-DIMENSIONAL INVERSE SCALAR PROBLEM OF DIFFRACTION BY AN INHOMOGENEOUS OBSTACLE
WITH A PIECEWISE CONTINUOUS REFRACTIVE INDEX 

Smirnov Yuriy Gennad'evich, Doctor of physical and mathematical sciences, professor, head of sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), E-mail: mmm@pnzgu.ru
Tsupak Aleksey Aleksandrovich, Candidate of physical and mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), E-mail: mmm@pnzgu.ru 

517.968, 517.983.37 

10.21685/2072-3040-2018-3-1 

Background. The aim of this work is theoretical study of the two-dimensional inverse scalar problem of diffraction by an inhomogeneous obstacle characterized with a piecewise continuous refractive index.
Material and methods. The original boundary value problem in the quasiclassical formulation is reduced to a system of integral equations; the properties of the latter system are studied using potential theory and Fourier transform.
Results. The integral formulation of the inverse diffraction problem is proposed; uniqueness of a piecewise constant solution to the Fredholm integral equation of the first type is established; novel two-step method for solving the inverse problem is proposed.
Conclusions. the proposed method and obtained results can be applied for solving two-dimensional problems of near-field tomography. 

two-dimensional inverse scattering problem, reconstruction of piecewise continuous refractive index, integral equations, uniqueness of solutions 

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