Solvability of the Integro-Differential Equation in the Problem of Wave Diffraction on a Junction of Rectangular Wavegui
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IAL DIFFERENTIAL EQUATIONS
Solvability of the Integro-Differential Equation in the Problem of Wave Diffraction on a Junction of Rectangular Waveguides A. S. Ilyinsky1∗ and Yu. G. Smirnov2,3∗∗ 1
Lomonosov Moscow State University, Moscow, 119991 Russia 2 Penza State University, Penza, 440026 Russia 3 Sirius University, Sochi, Krasnodar krai, 354340 Russia e-mail: ∗ [email protected], ∗∗ [email protected]
Received March 16, 2020; revised March 20, 2020; accepted May 14, 2020
Abstract— We study the problem of electromagnetic wave diffraction on a junction of two rectangular waveguides. The boundary value problem for the system of Maxwell equations is reduced to a vector pseudodifferential equation in special Sobolev spaces. Sufficient conditions are obtained for the existence of a unique solution of the boundary value problem and the integro-differential equation. DOI: 10.1134/S0012266120080078
INTRODUCTION The problem of electromagnetic wave diffraction on a junction of two rectangular waveguides is fundamental when designing the ducts of microwave devices [1]. The numerical solution of this problem, important in practical applications, has been known for a long time. However, a rigorous mathematical justification for the application of numerical methods has been missing, because this problem is reduced to a nonelliptic vector integro-differential equation along the junction aperture, and this equation is difficult to study by traditional methods. In the present paper, we study this equation by the method of pseudodifferential operators acting in special vector Sobolev spaces. Earlier, this approach permitted obtaining some results on the solvability of the corresponding integro-differential equations for the diaphragm junction of two half-spaces, a half-space and a layer, and a half-space and a rectangular waveguide [2–4]. Then this approach has successfully been applied to the analysis of other problems of the kind [5–9]. In this paper, we consider the electrodynamic problem of diffraction by a junction of two rectangular waveguides. The simpler problem of diaphragm junction of waveguides was studied in the paper [2]. In that case, the kernel of the integro-differential equation has a singularity only for coinciding arguments and is easier to study. We give a rigorous statement of the problem of diffraction by a junction of two rectangular waveguides. The electrodynamic parameters of the waveguides may differ. This is a three-dimensional vector problem that cannot be reduced to the scalar case. We provide a theorem on the uniqueness of the solution of the diffraction problem. We present Green’s functions for the waveguide domain, which is a semi-infinite rectangular cylinder. The singularity of Green’s functions is isolated. The singularity is isolated completely in the closed domain (including the boundary); this singularity is different for different domains. By introducing vector potentials and by using Green’s function, we reduce the diffraction problem for the system of Maxwell equations to a vector integro-di
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