MARTIN R. BARONE, DAVID A. CAULK; SPECIAL BOUNDARY INTEGRAL EQUATIONS FOR APPROXIMATE SOLUTION OF LAPLACE'S EQUATION IN TWO-DIMENSIONAL REGIONS WITH CIRCULAR HOLES. Q J Mechanics Appl Math 1981; 34 (3): 265-286. doi: 10.1093/qjmam/34.3.265
Special boundary integral equations are developed for Laplace's equation in a general two-dimensional region with an arbitrary configuration of circular holes. First, the solution on the hole boundaries is represented by a finite sum of circular harmonics with unknown coefficients. The hole geometry is then directly exploited in a new set of integral equations with special kernel functions which independently ‘pick out’ these coefficients. Each new equation contains only one coefficient relating to that particular hole, and so the resulting system of equations for the unknown field on the hole boundaries is inherently well-conditioned. The level of approximation in these equations depends on the number of harmonics in the representation of the solution on the hole boundary. Equations corresponding to the lowest and next higher level of approximation are solved in closed form for the examples of a single hole and an infinite row of holes in a half-space. In the case of a single hole, comparison with the known exact solution shows excellent agreement. The solution for the infinite row of holes is a significant improvement over a previously available approximation.