An integral equation related to the Riemann problem for a simply connected region with corners

Abstract

We shall formulate a new integral equation related to the Riemann problem, a class of boundary value problems for analytic functions, on a simply connected region, bounded by a curve having a continuously turning tangent except possibly at a finite number of corners in the complex plane. The boundary data are assumed to be continuous. The solution to this problem may be characterized as a solution to a singular integral equation on the boundary. By using results of Hille and Muskhelishvili, the theory is extended to include boundaries with corners which are rarely used for numerical computations, mainly due to singularities and other difficulties, arising from the calculation of Cauchy-type contour integrals operating on singular functions around irregular shaped domains. The complex Dirichlet problem, which is a well-known classical boundary value problem, is a particular case of the Riemann problem. Swarztrauber derived an integral equation on the numerical solution for Dirichlet problem for a region of general shape. He used Picard iteration and obtained an iterative formula, then he wrote it in a form so that in numerical integrations the singularities were eliminated. Here, our propose is to present a direct method by extending Swarztrauber's results. In this paper, our new integral equation for Riemann problem with similar region is formulated based on these results. A numerical implementation of solving integral equation using the Picard iteration is suggested; presenting an iterative formula that will eliminate singularities during numerical integrations.