Partial Differential Equations

Analytical inversion of the operator matrix for the problem of diffraction by a cylindrical segment in Sobolev spaces

S. I. Èminov

Yaroslav-the-Wise Novgorod State University

Abstract: A vector problem of electromagnetic-wave diffraction by a cylinder is described by a system of two two-dimensional integro-differential equations. After expanding the unknown functions and the right-hand sides in Fourier series, the problem reduces to systems of one-dimensional equations. Analytical inversion of the principal operator of one-dimensional systems in Sobolev spaces is considered. Theorems on the boundedness and bounded invertibility of the principal operator are proved. The inverse operator is represented by series and in closed form: the elements of the inverse matrix are integral or integro-differential operators.

Key words: diffraction, cylinder, operator matrix, Sobolev spaces, Lax–Milgram theorem, integral operator, singular operator, integro-differential operator, inverse matrix.

UDC: 517.927

Received: 15.01.2020
Revised: 15.01.2020
Accepted: 18.11.2020

DOI: 10.31857/S0044466921030054

 English version: Computational Mathematics and Mathematical Physics, 2021, 61:3, 424–430

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