Equivalence between the Trefftz method and the method of fundamental solutions for Green’s function of concentric spheres using the addition theorem and image concept

Abstract

Following the success of the mathematical equivalence between the Trefftz method and the method of fundamental solutions for the annular Green's function, we extend to solve the Green's function of 3-D problems in this paper. The Green's function of the concentric sphere is first derived by using the image method which can be seen as a special case of method of fundamental solutions. Fixed-fixed boundary conditions are considered. Also, the Trefftz method is employed to derive the analytical solution by using the T-complete sets. By employing the addition theorem, both solutions are found to be mathematically equivalent when the number of Trefftz bases and the number of image points are both infinite. In the successive image process, the final two images freeze at the origin and infinity, where their singularity strengths can be analytically and numerically determined in a consistent manner. The agreement among the three results, including two analytical solutions by using the Trefftz method and the image method, and one numerical solution by using the conventional MFS is observed.