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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).
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    Analysis of Hamiltonian boundary value problems and symplectic integration: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand
    (Massey University, 2020) Offen, Christian
    Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis (a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived. (b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem. (c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for $A$-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations. (d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers' equation, KdV, fluid equations,...) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation. (e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed.
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    The analysis of fragmentation type equation for special division kernels : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, School of Fundamental Sciences, New Zealand
    (Massey University, 2020) Almalki, Adel Ahmed
    The growth fragmentation equation is a linear integro-differential equation describing the evolution of cohorts that grow, divide and die or disappear in the course of time. The general formula is of first or second order, depending whether the growth process is deterministic or stochastic, respectively. We focus on a particular choice of division kernel that models size-structured cell cohorts which divide into daughter cells of equal size. This problem reduces to an initial-boundary value type that involves a modified Fokker-Planck equation with an advanced functional term. There are no general techniques for solving these problems. The constant growth rate case has been studied by a number of researchers. In particular, it was shown that the limiting solutions converge to a special solution, the separable solution. We consider the case when the growth rate is linear and deterministic. This problem can be solved analytically for monomial splitting rates. We show that the long time dynamics for this case differ markedly from the constant growth rate case. Specifically, the solutions approach a time dependent attracting solution that is periodic in time. The qualitative features of solutions differ when the splitting rate is constant. There are two cases. The first is when the growth rate is deterministic; the second is when the growth rate is stochastic. This case involves a constant dispersion term. In both cases, the problem can be solved directly, and the classic properties of solutions can be adapted from the previous case (with non constant splitting rate). The main distinct trait is that there is no long time attracting solution in $L^1$ for probability distribution initial data. (This result in the dispersive case follows, provided the parameters $g$, $b$ and $\alpha$ satisfy a certain inequality.) The long time asymptotic behaviour of solutions proves to be formidable to evaluate analytically for these cases. We use numerical methods to elucidate possible behaviour and examine the influence of the dispersion term. We find numerical evidence that the dispersion term plays a prominent r\^{o}le as a smoothing effect on the oscillatory behaviour, spatially and time wise, encountered in the dispersion free examples with exponential growth.
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    Mathematical modelling of fluid flow and heat and pollutant transport in a porous medium with embedded objects : a thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy (PhD) in Mathematics, Institute of Natural and Mathematical Sciences, Massey University, Albany, New Zealand
    (Massey University, 2018) Kubra, Khadija Tul
    How does heat and/or pollutant transfer from objects embedded in the ground depend on their size, shape and burial depth, and how does the dispersion of heat and/or pollution in groundwater aquifers depend on the soil properties, the speed of the groundwater flow, etc.? In detail, the aims of present study are: - To investigate how the size, shape and position of an object or set of solid or partially pervious objects, e.g., fluid tanks, pipes, etc., embedded in a porous medium affect the local speed and shape of the flow. - If heat is ejected from the solid objects e.g., fuel storage cylindrical tanks, radioactive waste reservoirs in deep geological formations, etc., and/or a pollutant is released from, or removed by, the pervious object, e.g., septic tanks, disposal of drums of contaminants, etc., how does the subsequent dispersal through a groundwater aquifer depend on the various parameters involved (e.g., the object size, object's burial depth, perviousness of the object, the aquifer's depth, the fluid flow rates, etc.)? - What is the effect of the non-homogeneity in matrix properties (e.g. permeability or hydraulic conductivity) on fluid flow, pollutant and heat transport rates? This study pursues answers to these questions. The porous medium fluid flow equations, and the advection-dispersion equations that model the heat and/or species transport, have coeffients that depend mainly on depth. Generally, analytic solutions are not possible. In order to investigate the effects of various objects of different shapes embedded in a porous medium, I have developed numerical algorithms and used some special mathematical techniques for two-dimensional models, namely conformal mappings within the framework of complex analysis. The velocity potential and (2-D) stream function satisfy Laplace's equation. Central and one-sided finite difference methods are applied to solve this equation subject to a chosen combination of constant-head or constant-flux boundary conditions. Results are discussed for various embedded shapes in homogeneous and layered groundwater aquifers. A Matlab command "contour" is used to depict the streamlines and equipotential lines, and the resulting temperature or pollutant concentrations. Steady-state and time-dependent forced convection heat/pollutant transfer from some cylinders embedded in groundwater are explored numerically using finite difference methods. The results show that the size, shape, position, perviousness and burial depth of the cylinder affect the pressure drop, as well as the pollutant and/or heat transfer. Moreover advection and dispersion depend on the permeability structure and the fluid speed.
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    An initial-boundary value problem arising in cell population growth modelling : a thesis submitted in partial fulfillment of the requirement for the degree of Master of Science in Mathematics at Massey University, (Manawatu) Institute of Fundamental Sciences, Mathematics
    (Massey University, 2012) Almalki, Adel
    A partial differential equation modelling cell populations undergoing growth and division characterised by size is studied. This equation is a special case of the fragmentation equation studied by Michel et al. [5] with no dispersion term, and the problem is of the initial-boundary value type. Eigenfunction solutions are derived for separable solutions and related properties such as spectrum, uniqueness and unimodality are investigated. We show that the spectrum is continuous and that the decay of the eigenfunctions is exponential at a critical eigenvalue and algebraic otherwise. The existence of a fast decay general solution n(x; t) is then established. The problem can be solved analytically, and it is shown that the solution is unique and smooth. The solution properties are illustrated with some numerical simulations. Finally, the role of exponential decaying eigenfunction solutions is interpreted from the standpoint of the general solution. The asymptotic behaviour as t ! 1 of the general solution is examined. Slow decay eigenfunction solutions are briefly discussed, but their mathematical role remains to be explored.