The Riemann–Hilbert problem and the generalized Neumann kernel

Abstract

This paper presents two new Fredholm integral equations associated to the interior and the exterior Riemann–Hilbert problems in simply connected regions with smooth boundaries. The kernel of these integral equations is the generalized Neumann kernel. The solvability of the integral equations depends on whether λ = ± 1 are eigenvalues of the kernel which in turn depends on the index of the Riemann–Hilbert problem. The complete discussion of the solvability of the integral equations with the generalized Neumann kernel is presented. The integral equations can be used effectively to solve numerically the Riemann–Hilbert problems. The case of non-uniquely solvable Riemann–Hilbert problems is treated by imposing additional constraints to get a uniquely solvable problem. Fredholm integral equations with generalized Neumann kernels are also derived for the problem of the interior and the exterior harmonic conjugate functions. As applications, we study the problem of conformal mapping to a nearby region and extend Wegmann's iterative method to general regions. Numerical examples reveal that the present method offers an effective solution technique for the Riemann–Hilbert problems when the boundaries are sufficiently smooth.