Bipolar coordinates, image method and the method of fundamental solutions for Green's functions of Laplace problems containing circular boundaries

Abstract

Green’s functions of Laplace problems containing circular boundaries are solved by using analytical and semi-analytical approaches. For the analytical solution, we derive the Green’s function using the bipolar coordinates. Based on the semi-analytical approach of image method, it is interesting to find that the two frozen images for the eccentric annulus using the image method are located on the two foci in the bipolar coordinates. This finding also occurs for the cases of a half plane with a circular hole and an infinite plane containing two circular holes. The image method can be seen as a special case of the method of fundamental solutions, which only at most four unknown strengths are required to be determined. The optimal locations of sources in the method of fundamental solutions can be captured using the image method and they are dependent on the source location and the geometry of problems. Three illustrative examples were demonstrated to verify this point. Results are satisfactory.