Verification of boundary integral equation for conformal mapping of doubly connected regions onto a disk with a slit

Abstract

In this study, we discussed a new Fredholm integral equation of the second kind with classical Neumann kernel associated to ????, where ? is a conformal mapping of bounded multiply connected regions onto a disk with slit domain. The boundary integral equation is constructed from a boundary relationship satisfied by a function that is analytic on a multiply connected region. The boundary integral equation is linear and does not contain any unknown radii. For numerical verification, we parameterized and discretized the integral equation by using the Nyström’s method with trapezoidal rule. Five test bounded doubly connected regions are chosen to verify the new boundary integral equation using the exact mapping functions. The five test regions are annulus, circular frame, frame of Limacon, elliptic frame and frame of Cassini’s oval.