ORCA
Online Research @ CardiffÂ

# A study of differential and integral operators in linear viscoelasticity

 Alzahrani, Faris 2013. A study of differential and integral operators in linear viscoelasticity. PhD Thesis, Cardiff University. Item availability restricted.

 Preview
PDF - Accepted Post-Print Version
PDF - Supplemental Material
Restricted to Repository staff only

## Abstract

This thesis identifies and explores a link between the theory of linear viscoelasticity and the spectral theory of Sturm-Liouville problems. The thesis is divided into five chapters. Chapter 1 gives a brief account of the relevant parts of the theory of linear viscoelasticity and lays the foundation for making the link with spectral theory. Chapter 2 is concerned with the construction of approximate Dirichlet series for completely monotonic functions. The chapter introduces various connections between non-negative measures, orthogonal polynomials, moment problems, and the Stieltjes continued fraction. Several interlacing properties for discrete relaxation and retardation times are also proved. The link between linear viscoelasticity and spectral theory is studied in detail in Chapter 3. The stepwise spectral functions associated with some elementary viscoelastic models are derived and their Sturm-Liouville potentials are explicitly found by using the Gelfand-Levitan method for inverse spectral problems. Chapter 4 presents a new family of exact solutions to the nonlinear integrodifferential A-equation, which is the main equation in a recent method proposed by Barry Simon for solving inverse spectral problems. Starting from the A-amplitude A(t) = A(t, 0) which is determined by the spectral function, the solution A(t, x) of the A-equation identifies the potential q(x) as A(0, x). Finally, Chapter 5 deals with two numerical approaches for solving an inverse spectral problem with a viscoelastic continuous spectral function. In the first approach, the A-equation is solved by reducing it to a system of Riccati equations using expansions in terms of shifted Chebyshev polynomials. In the second approach, the spectral function is approximated by stepwise spectral functions whose potentials, obtained using the Gelfand-Levitan method, serve as approximations for the underlying potential

Item Type: Thesis (PhD) Unpublished Mathematics Q Science > QA Mathematics 30 March 2016 19 Mar 2016 23:28 http://orca.cf.ac.uk/id/eprint/53333

### Actions (repository staff only)

 Edit Item