Khiem V. Ngo
{"title":"An Approach of Eigenvalue Perturbation Theory","authors":"Khiem V. Ngo","doi":"10.1002/anac.200410028","DOIUrl":"10.1002/anac.200410028","url":null,"abstract":"<p>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 (<i>A</i> + <i>εB</i>)<i>φ</i>(<i>ε</i>) = <i>λ</i>(<i>ε</i>)<i>φ</i>(<i>ε</i>) where <i>A</i> and <i>B</i> are <i>n</i> × <i>n</i> matrices, <i>ε</i> is a parameter, <i>λ</i>(<i>ε</i>) is an eigenvalue, and <i>φ</i>(<i>ε</i>) is the corresponding eigenvector. In working with perturbation theory, we assume that the eigenvalue <i>λ</i>(<i>ε</i>) 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 <i>λ</i>(<i>ε</i>). 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 <i>u</i><sub><i>tt</i></sub> = <i>c</i><sup>2</sup>Δ<i>u</i> with a fixed boundary Γ and is subjected to the boundary condition <i>u</i> = 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)</p>","PeriodicalId":100108,"journal":{"name":"Applied Numerical Analysis & Computational Mathematics","volume":"2 1","pages":"108-125"},"PeriodicalIF":0.0,"publicationDate":"2005-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/anac.200410028","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74053276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 16