Title page for ETD etd-07072011-143638


Document Type Master's Dissertation
Author Civin, Damon Jay
URN etd-07072011-143638
Document Title Riesz bases and the series representation of solutions of linear partial differential equations
Degree MSc
Department Mathematics and Applied Mathematics
Supervisor
Advisor Name Title
Prof N F J van Rensburg Supervisor
Keywords
  • linear vibration
  • generator
Date 2011-04-05
Availability restricted
Abstract
The vibration of an elastic body (or system of elastic bodies) is modelled by a partial differential equation or system of partial differential equations. Modal analysis is widely used in engineering to study vibration problems. This approach is based on the idea that eigenvalues and eigenvectors can be used to construct a series solution for a model problem. The validity of this series representation is investigated in this dissertation.

In the literature, linear vibration problems are converted to abstract first order linear differential equations. The series representation is valid if the generalized eigenvectors of the dynamics generator form a basis for the state space. However, the dynamics generator is non-normal when damping is present. There is no general spectral theory for such operators so special cases are considered in the literature. Moreover, there has been virtually no progress for problems that are two or three dimensional.

Various one dimensional linear vibration problems are considered in this dissertation, namely the wave equation with viscous, Kelvin-Voigt (material) or boundary damping and the Euler-Bernoulli beam with Kelvin-Voigt damping or boundary control.

The vibration problems considered are carefully formulated in the general framework of bilinear forms. The transition from a system of partial differential equations to an abstract first order differential equation is made rigorous. This is usually glossed over in the literature.

The general form of the dynamics generator of a linear vibration problem is found and it is proved that this dynamics generator is non-normal. In each case, the equivalence between weak and classical forms of the eigenvalue problems is considered and sufficient conditions for existence are found. For the wave equation with constant viscous damping, a Riesz basis is explicitly constructed and used to prove that the the energy of the solution decays exponentially.

An introduction to the spectral theory of nonselfadjoint operators and a summary of the disparate methods and results used in the literature are given. Then the various approaches taken by different authors to prove the Riesz basis property are presented.

First, a method using quadratically close sequences is discussed. An abstract result on the Riesz basis property is proved and applied to a beam problem.

Next, methods involving operator pencils and Krein spaces are discussed and a theorem on the Riesz basis property (due to Jacob, Trunk and Winklmeier) is generalized and made more directly applicable.

The third method, which is based on the biorthogonality of the eigenvectors is then presented. This dissertation concludes with an overview of the literature studied and comments on the similarities and differences between the methods encountered.

2010, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria.

Please cite as follows:

Civin, DJ 2010, Riesz bases and the series representation of solutions of linear partial differential equations, MSc dissertation, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-07072011-143638 / >

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