A Numerical Comparison of Integral Equations of the First and Second Kind for Conformai Mapping

Two methods for computing numerical conformai mappings are compared. The first, due to Symm, uses a Fredholm integral equation of the first kind while the other, due to Lichtenstein, uses a Fredholm integral equation of the second kind. The two methods are tested on ellipses with different ratios of major to minor axes. The method based on the integral equation of the second kind is superior if the ratio is less than or equal to 2.5. The opposite is true if the ratio is greater than or equal to 10. Similar results are obtained for other regions. Introduction. Let D be a bounded simply-connected region in the plane with a smooth boundary curve L. Then it is well known that there is a conformai mapping /which takes D in a 1-1 fashion onto the unit disc If I < 1 in such a way that a given point z0 E D is carried into f(z0) = 0. Moreover, / is uniquely determined up to an arbitrary rotation of the disc, and /can be extended continuously to D + L. The determination of /is equivalent to the determination of the Green's function of D. Recently, Symm [12] suggested a method for conformai mapping which represents the regular part of the Green's function of D as a single-layer potential on L. This results in a Fredholm integral equation of the first kind with a logarithmic kernel. An algorithm based on this integral equation for computing the mapping numerically [12], as well as an improved version [5], compared favorably with numerical methods based upon orthogonal polynomials. The utility of integral equations of the first kind for the solution of numerical problems has been viewed with skepticism. Indeed, a search of the literature shows that, whenever a problem can be formulated both as an equation of the first kind and as an equation of the second kind, the latter is almost always chosen. Recent work, however, shows that equations of the first kind can be the basis for useful numerical procedures (see [7], [4], [10]). The feasibility of numerically solving the integral equation used in Symm's method depends upon the nature of the spectrum of the associated operator. In [11] Received July 18, 1973 AMS (MOS) subject classifications (1970). Primary 30A28; Secondary 6SE05.