Identifier

etd-11182013-090349

Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

It is well known that all flows in a state space O induce a semigroup of linear operators on an appropriately chosen vector space of functions (observables) from O into a vector space Z (observations). After choosing appropriate continuity assumptions on the flow, the associated semigroup will be strongly continuous and will have a linear, infinitesimal generator A. The purpose of this dissertation is to explore approximation methods for linear semigroups and/or Laplace transform inversion methods in order to reconstruct the flow starting with the linear generator A . In preparing for these investigations, we collect some of the essential approximation theorems of semigroup theory and improve a recent generalization of the Trotter-Kato Theorem due to McAllister, Neubrander, Riser, and Zhuang. Moreover, we show that rational Laplace transform inversions of order m are exact for all polynomials of degree less than m. We will demonstrate that the flow can be efficiently reconstructed whenever the generator A of the induced semigroup has a resolvent that can be efficiently computed or approximated. We demonstrate this for flows solving nonlinear first order ordinary differential equations x'(t) = a(x(t)), x(s)= w and the induced generator (Af)(s)=a(s)f'(s) and for flows solving non-autonomous linear first order ordinary differential equations u'(t) = a(t)u(t), u(s)= w and the induced generator(Af)(s) = f'(s)+a(s)f(s). As a by-product of our investigation, we find a numerically efficient way to compute the inverse of increasing real-valued functions. Finally, we explore whether linear semigroup approximation methods can be used efficiently to approximate solutions of non-autonomous Cauchy problems u'(t) = A(t)u(t), u(s) = x in terms of the generator (Af)(s) = f'(s) + A(s)f(s) of the induced linear operator semigroup. As we will see, the Lie-Trotter approach suggested by G. Nickel seems to be the only efficient way to find the solutions of the non-autonomous problems in terms of the semigroup generated by A.

Date

2013

Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

Committee Chair

Neubrander, Frank

DOI

10.31390/gradschool_dissertations.1086

Download

Share

COinS