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114 Angermann and Yatsyk = δ n1 i(nκ) 2 2Γ nκ + δ n3 i(nκ) 2 ×α(ξ) ∫ 2πδ −2πδ ∫ 2πδ exp(iΓ nκ |z − ξ|)α(ξ)U 2 (2κ; ξ)Ū(3κ; ξ)dξ exp(iΓ nκ |z − ξ|) 2Γ nκ −2πδ { } 1 3 U 3 (κ; ξ) + U 2 (2κ; ξ)Ū(κ; ξ) dξ + U inc (nκ; z), |z| ≤ 2πδ, n = 1, 2, 3. (11) Here U inc (nκ; z) = a inc nκ exp [−iΓ nκ (z − 2πδ)] . The following result can be proved. Theorem 1 Assume that ε (L) , α are piecewise continuous, bounded and that all the data κ, δ, ϕ, {a inc nκ } 3 n=1 , α, and ε(L) satisfy 3∑ 492γ max |z|≤2πδ |1−ε(L) (z)| ≤ 35, max |α(z)| (a inc mκ) 2 ≤ 1721−2√ 10 |z|≤2πδ 551368γ with γ := πδκ/ cos ϕ. Then the iteration ∫ 2πδ m=1 U s+1 (nκ; z) + i(nκ)2 exp(iΓ nκ |z − ξ|) 2Γ nκ −2πδ ×[1 − ε nκ (ξ, α(ξ), U s (κ; ξ), U s (2κ; ξ), U s (3κ; ξ))]U s+1 (nκ; ξ)dξ = δ n1 i(nκ) 2 2Γ nκ + δ n3 i(nκ) 2 2Γ nκ ∫ 2πδ −2πδ ∫ 2πδ exp(iΓ nκ |z − ξ|)α(ξ)U 2 s (2κ; ξ)Ūs(3κ; ξ)dξ exp(iΓ nκ |z − ξ|) −2πδ { } 1 ×α(ξ) 3 U s 3 (κ; ξ) + Us 2 (2κ; ξ)Ūs(κ; ξ) dξ + U inc (nκ; z), |z| ≤ 2πδ, n = 1, 2, 3, s = 0, 1, 2, . . . , converges for sufficiently small initial values {U 0 (nκ; ·)} 3 n=1 such that 3∑ max |U 0 (mκ; z)| 2 ≤ 1/(82γ max |α(z)|) to the unique solution |z|≤2πδ m=1 |z|≤2πδ of the system (11). 3. NUMERICAL INVESTIGATION OF THE NONLINEAR INTEGRAL EQUATIONS AND SPECTRAL PROBLEMS The application of suitable quadrature rules to the system of nonlinear integral Equation (11) as described in [4–6] leads to a system of
Progress In Electromagnetics Research B, Vol. 56, 2013 115 complex-valued nonlinear algebraic equations: (I − B κ (U κ , U 2κ , U 3κ ))U κ = C κ (U 2κ , U 3κ ) + U inc κ , (I − B 2κ (U κ , U 2κ , U 3κ ))U 2κ = U inc 2κ , (I − B 3κ (U κ , U 2κ , U 3κ ))U 3κ = C 3κ (U κ , U 2κ ) + U inc 3κ , (12) where we use a discrete set {z l } N l=1 of nodes such that −2πδ =: z 1 < z 2 < . . . < z l < . . . < z N =: 2πδ. U nκ := {U l (nκ)} N l=1 ≈ {U (nκ; z l )} N l=1 and B nκ(U κ , U 2κ , U 3κ ) are the matrices generated by the quadrature method. The right-hand side of (12) is defined by U inc nκ := { a inc nκ exp[−iΓ nκ (z l − 2πδ)] } N l=1 { , iκ 2 ∑ N C κ (U 2κ ,U 3κ ) := A m exp(iΓ κ |z l − z m |) 2Γ κ C 3κ (U κ ,U 2κ ) := m=1 ×α(z m )U 2 m(2κ)Ūm(3κ) { i(3κ) 2 2Γ 3κ N ∑ m=1 } N l=1 A m exp(iΓ 3κ |z l − z m |) 1 ] ×α(z m )[ } N 3 U m(κ) 3 + Um(2κ)Ūm(κ) 2 where the numbers A m are the coefficients determined by the quadrature rule. The solution of (12) can be found iteratively, where at each step a system of linearised nonlinear complex-valued algebraic equations is solved. The system of nonlinear integral Equation (11) can be linearised directly by freezing the permittivities ε nκ . The analytic continuation of these linearised nonlinear problems into the region of complex values of the frequency parameter allows us to switch to the analysis of spectral problems. That is, the eigen-frequencies and the eigen-fields of the homogeneous linear problems with an induced nonlinear permittivity are to be determined. Analogously as above but at the discrete level we obtain a set of independent systems of linear algebraic equations depending nonlinearly on the spectral parameter: , l=1 (I − B nκ (κ n ))U κn = 0, (13) where κ n ∈ Ω nκ ⊂ H nκ , at κ = κ inc , n = 1, 2, 3, Ω nκ are the discrete sets of eigen-frequencies and H nκ denote two-sheeted Riemann surfaces ,
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