Explicit Relations for Harmonic Potentials with the Density Defined at Two Parallel Segments

Abstract

Explicit relations for harmonic potentials are obtained when the density of the potentials is defined at two segments of parallel straight lines. These results can be used for testing numerical algorithms of calculating the harmonic potentials in boundary value problems outside open arcs in a plane. Moreover, explicit solutions to two systems of singular integral equations are obtained, and these solutions can be used for testing numerical methods for solving singular integral equations. Futhermore, explicit solutions to Neumann problem for the Laplace equation are obtained for specific functions in the boudary conditions.