Image method for the Green's functions of annulus and half-plane Laplace problems

Abstract

In this paper, the image method is employed to solve the Green's function for the annular, eccentric and half-plane Laplace problems. For the annular case, not only the image method but also the Trefftz method is employed to derive the analytical solution. Their mathematical equivalence is also examined by using the bridge of addition theorem. For the half-plane problem with a circular hole and eccentric cases, a semi-analytical approach by using the image concept is used to solve the problems. After determining the three constants which contain one constant term and two strengths of two frozen image points by matching boundary conditions, we can obtain the semi-analytical solutions. The results are compared well with analytical solutions or other numerical results. In addition, it is interesting to find that the image locations can be seen as the optimum location for the source in the method of fundamental solution.