In this thesis we consider the problem of recovering the relaxation spectrum from the storage and loss moduli. We invert an integral equation using Fourier transforms. Recovering the relaxation spectrum is an inverse, ill-posed problem and hence regularisation methods must be used to try and obtain the relaxation spectrum. We are particularily interested in establishing properties of the relaxation spectrum. We note from the literature that there are results of compact support for the relaxation spectrum; we review to what extent and in what sense, these results are valid. We consider the methods used in the literature and demonstrate their strengths and weaknesses, supplying some missing details. We demonstrate in chapter 3 the difficulty in obtaining an interval of compact support for the relaxation spectrum and in the remainder of chapter 3 and chapter 4 we prove results of non-compactness of support for non-trivial relaxation spectra. Our settings are square integrable functions in chapter 3, and Schwartz distributions in chapter 4; we make use of Paley-Wiener theorems. These are important results since they contradict results in the literature that we review in chapter 2. We are able to demonstrate, using examples and via direct calculations, that the relaxation spectrum becomes insignificant outside some closed interval. With regards to numerical computations, this could be considered as a weak form of compact support. We call this essential numerical support; this may be a useful concept for the practical rheologist.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:575404 |
| Date | January 2012 |
| Creators | Whittle Gruffudd, Hannah Rebecca |
| Contributors | Douglas, Robert ; Mishuris, Gennady |
| Publisher | Aberystwyth University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/2160/f2b30f89-dc62-4038-83c9-1857eca8a2b5 |
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