Numerical conformal mapping via a boundary integral equation with the adjoint generalized Neumann kernel

This paper presents a new uniquely solvable boundary integral equation for computing the conformal mapping, its derivative and its inverse from bounded multiply connected regions onto the five classical canonical slit regions. The integral equation is derived by reformulating the conformal mapping as an adjoint Riemann-Hilbert problem. From the adjoint Riemann-Hilbert problem, we derive a boundary integral equation with the adjoint generalized Neumann kernel for the derivative of the boundary correspondence function $\theta'$. Only the right-hand side of the integral equation is different from a canonical region to another. The function $\theta'$ is integrated to obtain the boundary correspondence function $\theta$. The integration constants as well as the parameters of the canonical region are computed using the same uniquely solvable integral equation. A numerical example is presented to illustrate the accuracy of the proposed method.