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: ∇ 2 w + k 2 w = 0 , x ∈ Ω ⊂ R 2 , 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.