The numerical solution of well posed Stefan problems in a two dimensional region are considered using a boundary integral technique. The numerical method is an extension of that used in previous work in that the boundary integral formulation takes account of heat flow both ahead of and behind the phase change front. This allows more realistic problems to be considered. Furthermore, it is found that when parameter values appropriate for water are used, the previously applied routine, based on Newton's method, for determining the location of the phase change front, is unstable. This is overcome by using a dissection based method for these parameter values. This numerical formulation is found to have a number advantages over finite difference and finite element techniques. For example, complex boundaries can be easily considered and the discretisation of the Stefan condition is not required. Numerical solutions of the Stefan problem are found for different parameter values and, more specifically, the freezing of water is considered. Employing a model of crystal formation, the numerical method is applied to predict the size of the crystals in the crystalline micro-structure that is formed when a material freezes. The predictions of this model are compared against experimental results and it is found that they are in good qualitative agreement. To obtain a more accurate model of the freezing of a liquid, the numerical method is extended to include the fluid motion in the closed region ahead of the phase change front. A numerical procedure is outlined for dealing with genuine two-phase problems, using a different approach in each phase. The fluid flow problem in the liquid phase of the material is solved using a time dependent finite difference method on a non-uniform mesh, whereas the previously derived boundary integral method is used to determine the temperature distribution in the solid phase. The numerical scheme in the liquid phase is found to be second order in space and time.

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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.

This thesis applies rearrangement and optimal mass transfer theory to weather forecast error decomposition. Errors in weather forecasting are often due to displacement of key features; conventional error scores do not necessarily favour good forecasts, nor are they descriptive of how the forecast failed. We study forecast error decomposition, where error is split into an error due to displacement and an error due to differences in qualitative features. In its simple formulation, we seek re-arrangements of the forecast which are a best fit to the actual data, and then find the “least kinetic energy” of a notional velocity transporting the forecast to a best fit. In mathematical terms, we are characterising those elements of a set of rearrangements which are closest (in the sense of L2) to a prescribed square integrable function, and seeking the least 2-Wasserstein distance squared between the forecast and the closest displaced forecasts. We demonstrate that there are closest rearrangements, and characterise this set; the best fitting rearrangements are determined up to rearrangement on the level sets of positive size of the prescribed function. Displacement error is calculated by finding the minimum value of an optimal mass transfer problem; we review previous work, demonstrating the connection with transport of the forecast to the best fit. A problem with the simple formulation of forecast error decomposition is that because the qualitative features error is taken first, an error in qualitative features may be penalise as a large displacement error. We conclude this thesis by considering a formulation which minimises both errors simultaneously.

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