An Approach of Eigenvalue Perturbation Theory

Abstract

This paper presents an approach of eigenvalue perturbation theory, which frequently arises in engineering and physical science. In particular, the problem of interest is an eigenvalue problem of the form (A + εB)φ(ε) = λ(ε)φ(ε) where A and B are n × n matrices, ε is a parameter, λ(ε) is an eigenvalue, and φ(ε) is the corresponding eigenvector. In working with perturbation theory, we assume that the eigenvalue λ(ε) has a power series expansion. As such, a large effort presented in this paper involves the derivation of formulas for the power series coefficients, which are used to approximate λ(ε). In the process, the analysis requires some basic background of complex function theory. The rest of this paper presents an application of this approach to a common problem in engineering, namely, the vibration of a square membrane under the effect of a small perturbation, which results in a shape of a trapezoid. The displacement of the membrane of this particular shape is described by the differential equation u_tt = c²Δu with a fixed boundary Γ and is subjected to the boundary condition u = 0 on Γ. While the solution of the unperturbed hyperbolic problem of this type is well known and easy to find, it becomes quite difficult when the domain is perturbed, giving rise to a slightly different shape other than the original standard shapes, such as squares, rectangles, or circles. This paper addresses one of these aspects in which the domain results in a shape of a trapezoid. The approach should apply to other shapes as well. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)