Title page for ETD etd-07112012-204148


Type of Document Dissertation
Author Windsperger, Lee Gregory
URN etd-07112012-204148
Title Operational Methods for Evolution Equations
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Neubrander, Frank Committee Chair
Delzell, Charles Committee Member
Estrada, Ricardo Committee Member
Litherland, Richard Committee Member
Shipman, Stephen Committee Member
Wang, Ying Dean's Representative
Keywords
  • Evolution Equations
  • Laplace Transform
  • Rational Approximation of Semigroups
  • Numerical Approximation
Date of Defense 2012-07-06
Availability unrestricted
Abstract

This dissertation refines and further develops numerical methods for the inversion of the classical Laplace transform and explores the effectiveness of these methods when applied (a) to an asymptotic generalization of the Laplace transform for generalized functions and (b) to the numerical approximation of solutions of ill-posed evolution equations (e.g. backwards in time problems).

Chapter 1 of the dissertation reviews some of the key features of asymptotic Laplace transform theory and its application to evolution equations. Although some of the statements and results contain slight modifications and improvements, the material presented in Chapter 1 is known to the experts in the field. The main contributions of this work is in Chapter 2 where an attempt is made to help clarify and determine the size of the constant in the celebrated Hersh-Kato and Brenner-Thomče approximation theorem of semigroup theory. In particular, by improving an earlier estimate, we are able to show that matrix semigroups etA can be approximated "without scaling and squaring" in terms of the resolvent R(λ,A)=I -A)-1 of the generating matrix A (see Theorem 2.3.2). Also, our estimate of the Brenner-Thomče constant given in Section 2.4 improves earlier estimates given by Neubrander, Özer, and Sandmaier in [28]. The techniques used in Section 2.4 open the door to Theorem 2.5.1, a first attempt to lift the matrix result (Theorem 2.3.2) to the general semigroup setting. Finally, in (2.31) we present a new approach on how to approximate the continuous representatives f=k*u of a generalized function u in terms of its Laplace transform .

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