Analytical Approach to Boundary Integral Formulation for the Laplace/Poisson Equations in Circular Regions

A use of boundary integral equation (BIE) in solving the boundary problems in such 2D domains like circles and rings is considered in the paper. By expanding the excitation and the solution into the Fourier series, the problem is reduced to purely algebraic for each angular harmonic. The fundamental solutions for each harmonic are determined for the Laplace equation. Examples of use of the reduced BIE are given. Analysis for various combinations of boundary conditions is performed. One of the consequences is explaining the reason why the conventional boundary element method (BEM) crashes when solving the Laplace equation in a circle of unit radius (degenerate scale problem). As a kind of by-product, some definite integrals were found.