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Subject: search.filters.subject.490103 Calculus of variations, mathematical aspects of systems theory and control theory

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    Generalised diffusion equations for anomalous diffusion in polymer networks : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics, School of Natural Sciences, Massey University, Manawatu, New Zealand
    (Massey University, 2022) Cleland, Josiah David
    Fractional and generalised derivative equations have proven to be a powerful tool in the modelling of anomalous diffusion within complex fluids. The suitability of these equations arises from to their capacity to include memory effects prevalent in the fluid, as well as allowing for the inclusion of external forces and relevant boundary values. Fractional derivative equations may be derived from underlying continuous-time random-walk models, however, the fractional derivative equations (and their generalisations) are simpler to deal with. This thesis investigated the ability of fractional derivative equations (and their generalisations) to model long timescale anomalous diffusion phenomena observed in some visco-elastic polymer networks. Recent work has suggested that within certain physical polymer networks there is a tendency for internal stresses to continuously build and dissipate. This phenomenon manifests itself within recordings of probe-particle mean squared displacements at long time scales. The dynamic behaviour of these networks parallels behaviour observed in earthquakes, earning the phenomena the name gel or cyto quakes. This link suggests that statistical features involved in other stress driven events may provide insight into the modeling of gel quakes. Both temporal and spatial considerations relevant to these quaking systems will be outlined and modelled within this work.
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    Minimisation of mean exponential distortions and Teichmüller theory : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
    (Massey University, 2019) Yao, Cong
    This thesis studies the Cauchy boundary value problem of minimising exponential integral averages of mappings of finite distortion. Direct methods in calculus of variations provide existence theorems and we derive the Euler-Lagrange equations for minimisers of ∫D exp(pK(z, f)) dz for mappings of finite distortion f : D → D with prescribed boundary values. However, surprisingly, for these functionals some apriori regularity is needed before we can discuss these equations. We show by example how this can happen. We construct a mapping f : D → D with exponentially integrable distortion to exponent p which cannot perturbed by any diffeomorphism and still remain exponentially integrable with exponent p. Once enough apriori regularity is assumed for instance if a minimiser is locally quasiconformal, that is if the distortion function K(z, f) is locally bounded, then we use these equations to improve the regularity of the minimisers. In particular, we find that minimisers with locally bounded distortions are diffeomorphisms. Then we analyse the two extreme cases (1) p → 0 and (2) p → ∞. In this way we see the p-exponential problem connects the L¹ finite distortion problem, which is closely related to the classical harmonic theory in case (1), and to the Teichmüller problem, which promoted the development of quasiconformal mappings, in case (2).