A Rectangular Quadrature Method for Logarithmically Singular Integral Equations of the First Kind

Interpolation on Arbitrary Regions in the Complex Plane

“…The author also thanks Dr. Hrycak for providing the FMM-based fast interpolation routine and the referees of this paper for pointing out Refs. [3,12,18,19,23].…”

“…Such interpolation points can be obtained via the solution of an integral equation of the first kind, which is also known as the modified Symm's integral equation and has received considerably attention during the last 15 years (see [18,19,23]) in the construction of exterior conformal mappings (for other methods of solving integral equations of the first kind; see [3,12]) However, the classical solution requires order N 3 operations, where N is the number of nodes in the discretization of the given curve.…”

Fully-Discrete Collocation Methods for an Integral Equation of the First Kind

“…Our approach is quite different, being based on the theory developed in [8], and is thus related to the qualocation method, see Chandler and Sloan [5] and the survey paper [14], and also to the fully-discrete Galerkin methods in [10] and [11]. There are many other fully-discrete schemes for the log-kernel equation (1.1), such as the one studied in a recent paper of Bialecki and Yan [3]. These schemes, however, have no direct interpretation as full discretizations of the standard splinecollocation method.…”