Novel method of fundamental solutions formulation for polyharmonic BVPs

Abstract

We propose a novel method of fundamental solutions (MFS) formulation for solving boundary value problems (BVPs) governed by the polyharmonic equation ΔNu=0,N∈N∖{1}, in Ω⊂Rd,d=2,3. The solution is approximated by a linear combination of the fundamental solution of the operator ΔN and its first N−1 derivatives along the outward normal vector to the MFS pseudo–boundary. The optimal position of the pseudo–boundary on which the source points are placed is found using the effective condition number technique. Moreover, the proposed technique, when applied to polyharmonic BVPs in radially symmetric domains, lends itself to the application of matrix decomposition algorithms. The effectiveness of the method is demonstrated on several numerical examples.