The method of external excitation for solving Laplace singular eigenvalue problems

In this paper a new numerical technique for Laplace eigenvalue problems in the plane: ∇2w+k2w=0,x∈Ω⊂R2,B[w]=0,x∈∂Ω is presented. We consider the case when the solution domain has boundary singularities like a reentrant corner, or an abrupt change in the boundary conditions. The method is based on mathematically modelling of physical response of a system to excitation over a range of frequencies. The response amplitudes are then used to determine the resonant frequencies. We use the local Fourier-Bessel basis functions to describe the behaviour of the solution near the singular point. The results of the numerical experiments justifying the method are presented. In particular, the L-shaped domain and the cracked beam eigenvalue problems are considered.