Image solutions for boundary value problems without sources

In this paper, we employ the image method to solve boundary value problems in domains containing circular or spherical shaped boundaries free of sources. two and threeD problems as well as symmetric and anti-symmetric cases are considered. By treating the image method as a special case of method of fundamental solutions, only at most four unknown strengths, distributed at the center, two locations of frozen images and one free constant, need to be determined. Besides, the optimal locations of sources are determined. For the symmetric and anti-symmetric cases, only two coefficients are required to match the two boundary conditions. The convergence rate versus number of image group is numerically performed. The differences of the image solutions between 2D and 3D problems are addressed. It is found that the 2D solution in terms of the bipolar coordinates is mathematically equivalent to that of the simplest MFS with only two sources and one free constant. Finally, several examples are demonstrated to see the validity of the image method for boundary value problems.