Stationary Oscillations of Elastic Plates A Boundary Integral Equati
Elliptic partial differential equations are important for approaching many problems in mathematical physics, and boundary integral methods play a significant role in their solution. This monograph investigates the latter as they arise in the theory c
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Stationary Oscillations of Elastic Plates A Boundary Integral Equation Analysis
Gavin R. Thomson A.C.C.A. 89 Hydepark Street Glasgow, G3 8BW UK [email protected]
Christian Constanda The Charles W. Oliphant Professor of Mathematical Sciences Department of Mathematical and Computer Sciences The University of Tulsa 800 S. Tucker Drive Tulsa, OK 74104-9700 USA [email protected]
ISBN 978-0-8176-8240-8 e-ISBN 978-0-8176-8241-5 DOI 10.1007/978-0-8176-8241-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011931291 Mathematics Subject Classification (2010): 31A10, 74K20, 74H45 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com)
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Preface
Many important problems in mathematical physics can be modeled by means of elliptic partial differential equations or systems. Such equations arise in the study of, for example, steady-state heat conduction (the Laplace equation), acoustics (the Helmholtz equation), elasticity (the Lam´e system), and electromagnetism (the Maxwell system). An important tool for investigating boundary value problems associated with equations of this type is the boundary integral equation technique, which relies on the derivation of Fredholm or quasi-Fredholm integral equations over the boundary of the region of interest and leads to a very convenient representation of the solution. The kernels of the ensuing integral equations are expressed in terms of a two-point (scalar or matrix) function that is, in fact, a fundamental solution of the governing linear differential operator. Boundary integral equation methods are extremely useful for a variety of reasons. First, they reduce the problem from one involving an unbounded partial differential operator to one with an integral operator, making it much more appealing from an analytic perspective; second, the methods are very general in that they can be applied to any linear second-order elliptic boundary value problem with constant coefficients; and third, the methods are attractive from a numerical point of view because they yield closed-form solutions and, therefore, lend themselves readily to boundary element treatment. Boundary integral equation methods com
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