Justification of a Numerical Method for Solving Systems of Singular Integral Equations in Diffraction Grating Problems

摘要

It was shown in [1-4] that electromagnetic diffraction problems both for finite gratings consisting of finitely many thin ideally conducting strips and for multielement periodic gratings result in a singular integral equation of the first kind on the system of segments: 1/π ∫_L F(ξ)dξ/ξ-x + 1/π∫_L K(x,ξ)F(ξ)dξ = f(x), x ∈L (1) where f(x) x ∈ L-bar, and K(x, ξ), x ∈ L-bar, ξ ∈ L-bar, are smooth functions, L = ∪_(q=1)~m (a_q, b_q), - ∞ < a_1 < b_1 < … < a_m < b_m < +∞, and the function F(ξ), ξ ∈ L, is sought in the class of functions whose restrictions F_q(ξ) ≡ F(ξ), a_q < ξ < b_q, q = 1, …, m, to the interval (q_q, b_q) in accordance with the condition on the edge can be represented in the form F_q(ξ) = v_q(ξ)/[(ξ-a_q) (b_q - ξ)]~(1/2), q_q < ξ < b_q, where v_q(ξ), ξ ∈ [a_q, b_q], is a smooth function.