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Numerical conformal mapping of doubly connected regions via the Kerzman-Stein kernel. (English) Zbl 1149.30012

Authors’ abstract: Denote by G a ring domain with a smooth boundary Γ=Γ 0 Γ 1 and by A={w:μ<|w|<1} its conformal image. It is known that the mapping g,g(G)=A satisfies the integral equation

h(z)+ Γ A(z,w)h(w)|dw|+i(1-μ)T(z) ¯ Γ 2 h(w)[(w-z)g(w)] -1 dw ¯=0,(*)

where T(z) is the unit tangent vector, h(z)=g ' (z), H(w,z)=T(z)[2πi(z-w)] -1 , A(z,w)=H(w,z) ¯-H(w,z), Γ 2 =-Γ 1 ,zΓ 0 Γ 0 ,zΓ 1 . A(z,w) is the Kerzman-Stein kernel. It is smooth and skew-Hermitian. The equation (*) is separated into a system of two integral equations and another equation involving the modulus μ. The discritized integral equation leads to a system of nonlinear equations which is to be solved by an optimization method. An advantage of this approach is that it calculates the boundary correspondence functions and the modulus μ simultaneously. Some numerical examples are provided.

MSC:
30C30Numerical methods in conformal mapping theory
45G15Systems of nonlinear integral equations
65E05Numerical methods in complex analysis