Synthesis of Surface H-Polarized Currents on an Unclosed Cylindrical Surface

The article describes the inverse problem of diffraction of electromagnetic waves, finding surface H-polarized currents on an unclosed cylindrical surface according to a given radiation pattern. The work is based on modelling an operator equation with a small parameter. The operator is represented as the sum of a positive-definite, continuously invertible operator and a compact positive operator. The positive-definite operator exactly coincides with the main operator of the corresponding direct problem of diffraction of electromagnetic waves. Due to this fact, the solution to the simulated equation satisfies the necessary boundary conditions. And this is the novelty and difference of the approach developed in this work from the methods known in the scientific literature. We develop a theory of an operator equation with a small parameter and a numerical method based on Chebyshev polynomials with weights that take into account the behavior at the boundary. The efficiency of the numerical method is shown.

Keywords

inverse diffraction problem; equation with a small parameter; positive definite operator; completely continuous operator; Hilbert space.

References

1. Bakhrakh L.D., Kremenetskiy S.D. Sintez izluchayushchikh sistem. Teoriya i metody rascheta [Synthesis of Radiating Systems (Theory and Methods of Calculation)]. Moscow, Sovetskoe radio, 1974. (in Russian)
2. Zakharov E.V., Pimenov Yu.V. Chislennyy analiz difraktsii radiovoln [Numerical Analysis of Radio Wave Diffraction]. Moscow, Radio i svyaz', 1982. (in Russian)
3. Katsenelenbaum B.Z. Problemy approksimiruemosti elektromagnitnogo polya [Electromagnetic Field Approximability Problems]. Moscow, Nauka, 1996. (in Russian)
4. Ivanov V.K., Vasin V.V.,Tanana V.P. Teoriya lineynykh nekorrektnykh zadach i ee prilozheniya [The Theory of Linear Ill-Posed Problems and Its Applications]. Moscow, Nauka, 1978. (in Russian)
5. Sochilin A.V., Eminova V.S., Eminov I.S. [The Galerkin Method in the Problem of Diffraction of the H-Polarization on a Cylindrical Surface]. Non-linear World, 2014, no. 6, pp. 26-31. (in Russian)
6. Mikhlin S.G. Lineynye uravneniya v chastnykh proizvodnykh [Linear Partial Differential Equations]. Moscow, Nauka, 1977.
7. Eminova V.S., Eminov S.I. [Justification of the Galerkin Method for Hypersingular Equations]. Journal of Computational Mathematics and Mathematical Physics, 2016, vol. 56, no. 3, pp. 432-440. (in Russian)
8. Sakhnovich L.A. [Equations with a Difference Kernel on a Finite Interval]. Russian Mathematical Surveys, 1980, vol. 35, no. 4, pp. 69-129. (in Russian)
9. Rudin W. Functional Analysis. N.Y., McGRAW-Hill Book Company, 1973. \n10. Sukacheva T.G., Matveeva O.P. [Taylor Problem for the Zero-Order Model of an Incompressible Viscoelastic Fluid], Differential Equations, 2015, vol. 51, no. 6, pp. 783-791.