ON THE TRUE AND SPURIOUS EIGENVALUES BY USING THE REAL OR THE IMAGINARY-PART OF THE METHOD OF FUNDAMENTAL SOLUTIONS

Abstract

In this paper, the method of fundamental solutions (MFS) of real-part or imaginary-part kernels is employed to solve two-dimensional eigenproblems. The occurring mechanism of spurious eigenvalues for circular and elliptical membranes is examined. It is found that the spurious eigensolution using the MFS depends on the location of the fictitious boundary where the sources are distributed. By employing the singular value decomposition technique, the common left unitary vectors of the true eigenvalue for the single- and double-layer potential approaches are found while the common right unitary vectors of the spurious eigenvalue are obtained. Dirichlet and Neumann eigenproblems are both considered. True eigenvalues are dependent on the boundary condition while spurious eigenvalues are different in the different approach, single-layer or double-layer potential MFS. Two examples of circular and elliptical membranes are numerically demonstrated to see the validity of the present method and the results are compared well with the theoretical prediction.