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    Point and Lie Bäcklund symmetries of certain partial differential equations : a thesis presented in partial fulfilment of the requirements for the degree of MA in Mathematics at Massey University
    (Massey University, 1994) Pigeon, David Leslie
    The aim of this thesis is to: (1) Explore the use of differential forms in obtaining point and contact symmetries of particular partial differential equations (PDEs) and hence their corresponding similarity solutions. [1] and [4]. (2) Explore the generalized or Lie-Bäcklund symmetries of particular PDEs with particular reference to the Korteweg-de Vries-Burgers (KdVB) equation [3]. Finding point symmetries of a PDE H = 0 with independent variables (x1,x2 ) which we take to represent space and time and dependent variable (u) means finding the transformation group that takes the variables (x1, x2, u) to the system (x´1, x´2 , u´ ) and maps solutions of H = 0 into solutions of the same equation. The form of H = 0 remains invariant. The transformation group is usually expressed in terms of its infinitesimal generator (X) where using the tensor summation convention. X can be considered as a differential vector operator with components (ξ1 , ξ2 , η) operating in a three dimensional manifold (space) with coordinates (x1 , x2 , u). The invariance of H = 0 under the transformation group is expressed in terms of a suitable prolongation or extension of X (denoted by X(pr) ) to cover the effect of the transformations on the derivatives of u in H = 0. The invariance condition for H = 0 under the action of the transformation group is (Pr) [H] = 0 whenever H = 0. We consider x1 , x2 , u and the derivatives of u to be independent variables. In practical terms, finding point symmetries of H = 0 means finding the components (ξ1 , ξ2 , η) of the infinitesimal generator (X). There are two general methods for finding ξ1 , ξ2 η. [From Introduction] [NB: Mathematical/chemical formulae or equations have been omitted from the abstract due to website limitations. Please read the full text PDF file for a complete abstract.]
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    Symplectic integrators for vakonomic equations and for multi-Hamiltonian equations : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
    (Massey University, 2016) Wilkins, Matthew Colin
    Almost 200 years ago William Hamilton gave the world his reformulation of classical mechanics: the so-called Hamiltonian mechanics. By permitting a singular structure matrix, Mr Wilkins’ research extended this exalted theory to accommodate the Vakonomic equations, consequently allowing a solution to the sub-Riemannian geodesic and optimal control problems within this framework. The multi-Hamiltonian equation is an extension of Hamiltonian mechanics that appears in fields ranging from quantum mechanics to classical electrodynamics. Mr Wilkins’ research was conducted to the highest standards using numerical and theoretical proof and provided a stable, high-order multisymplectic numerical method for solving the multi-Hamiltonian equations where none previously existed. Our knowledge has increased because Hamiltonian mechanics has been extended to accommodate the Vakonomic equations and humanity now has a high-order multisymplectic numerical method for solving multi-Hamiltonian equations.
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    Travelling wave solutions in multisymplectic discretisations of wave equations : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand
    (Massey University, 2013) McDonald, Fleur Cordelia
    Symplectic integrators for Hamiltonian ODEs have been well studied over the years and a lot is known about these integrators. They preserve the symplecticity of the system which automatically ensures the preservation of other geometric properties of the system, such as a nearby Hamiltonian and periodic and quasiperiodic orbits. It is then natural to ask how this situation generalises to Hamiltonian PDEs, which leads us to the concept of multisymplectic integration. In this thesis we study the question of how well multisymplectic integrators capture the long-time dynamics of multi-Hamiltonian PDEs. We approach this question in two ways|numerically and through backward error analysis (BEA). As multi-Hamiltonian PDEs possess travelling wave solutions, we wish to see how well multisymplectic integrators preserve these types of solutions. We mainly use the leapfrog method applied to the nonlinear wave equation as our test problem and look for the preservation of periodic travelling waves. We call the resulting equation the discrete travelling wave equation. It cannot be solved exactly. Therefore, our analysis begins with numerically solving the discrete travelling wave equation for simpli ed nonlinearities. Next, we mov on to analysing periodic solution for a smooth nonlinearity. This results in the presence of resonances in the solutions for certain combinations of the parameters. Finally, we use backward error analysis to compare and back up our results from numerical analysis.
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    Resolving decomposition by blowing up points and quasiconformal harmonic extensions : 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, 2012) Dillon, Samuel Adam Kuakini
    In this thesis we consider two problems regarding mappings between various two-dimensional spaces with some constraint on their distortion. The first question concerns the use of mappings of finite distortion that blow up a point where the distortion is in some Lp class; in particular, we are interested in minimal solutions to the appropriate functional. We first prove some results concerning these minimal solutions for a given radially symmetric metric (in particular the Euclidean and hyperbolic metrics) by proving a theorem which states the conditions under which a minimizer exists, as well as providing lower bounds on the Lp-norm of the function. We then apply these results to the problem of resolving decompositions that arise in the study of Kleinian groups and the iteration of rational maps. Here we prove a result concerning for which values of p we can find a mapping of a particular form which shrinks the unit interval and whose inverse has distortion in the Lp space. The second is in regards to the Schoen conjecture, which expresses the hope that every quasisymmetric self-mapping of the unit circle extends to a homeomorphism of the disk which is both quasiconformal and harmonic with respect to the hyperbolic metric. The equation for a harmonic map between Riemann surfaces with given conformal structures is a nonlinear second order equation; one wishes to solve the associated boundary value problem. We show here that the existence question can be related to a nonlinear inhomogeneous Beltrami equation and discuss some of the consequences; this result holds in more generality for other conformal metrics as well.
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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.
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    Preservation of phase space structure in symplectic integration : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
    (Massey University, 2009) O'Neale, Dion Robert James
    This thesis concerns the study of geometric numerical integrators and how they preserve phase space structures of Hamiltonian ordinary differential equations. We examine the invariant sets of differential equations and investigate which numerical integrators preserve these sets, and under what conditions. We prove that when periodic orbits of Hamiltonian differential equations are discretized by a symplectic integrator they are preserved in the numerical solution when the integrator step size is not resonant with the frequency of the periodic orbit. The preservation of periodic orbits is the result of a more general theorem which proves preservation of lower dimensional invariant tori from dimension zero (fixed points) up to full dimension (the same as the number of degrees of freedom for the differential equation). The proof involves first embedding the numerical trajectory in a non-autonomous flow and then applying a KAM type theorem for flows to achieve the result. This avoids having to prove a KAM type theorem directly for the symplectic map which is generally difficult to do. We also numerically investigate the break up of periodic orbits when the integrator's step size is resonant with the frequency of the orbit. We study the performance of trigonometric integrators applied to highly oscillatory Hamiltonian differential equations with constant frequency. We show that such integrators may not be as practical as was first thought since they suffer from higher order resonances and can perform poorly at preserving various properties of the di fferential equation. We show that, despite not being intended for such systems, the midpoint rule performs no worse than many of the trigonometric integrators, and indeed, better than some. Lastly, we present a numerical study of a Hamiltonian system consisting of two magnetic moments in an applied magnetic field. We investigate the effect of both the choice of integrator and the choice of coordinate system on the numerical solutions of the system. We show that by a good choice of integrator (in this case the generalised leapfrog method) one can preserve phase space structures of the system without having to resort to a change of coordinates that introduce a coordinate singularity.
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    Optimal harvesting strategies for fisheries : a differential equations approach : a thesis presented in partial fulfillment of the requirement for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
    (Massey University, 2008) Suri, Ratneesh
    The purpose of fisheries management is to achieve a sustainable development of the activity, so that future generations can also benefit from the resource. However, the optimal harvesting strategy usually maximizes an economically important objective function formed by the harvester which can lead to the extinction of the resource population. Therefore, sustainability has been far more difficult to achieve than is commonly thought; fish populations are becoming increasingly limited and catches are declining due to overexploitation. The aim of this research is to determine an optimal harvesting strategy which fulfills the economic objective of the harvester while maintaining the population density over a pre-specified minimum viable level throughout the harvest. We develop and investigate the harvesting model in both deterministic and stochastic settings. We first employ the Expected Net Present Value approach and determine the optimal harvesting policy using various optimization techniques including optimal control theory and dynamic programming. Next we use real options theory, model fish harvesting as a real option, and compute the value of the harvesting opportunity which also yields the optimal harvesting strategy. We further extend the stochastic problem to include price elasticity of demand and present results for di¤erent values of the coefficient of elasticity.
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    Dynamics and numerics of generalised Euler equations : a thesis submitted to Massey University in partial fulfillment of the requirements for the degree of Ph.D. in Mathematics, Palmerston North, New Zealand
    (Massey University, 2008) Zhang, Xingyou
    This thesis is concerned with the well-posedness, dynamical properties and numerical treatment of the generalised Euler equations on the Bott-Virasoro group with respect to the general H[superscript]k metric , k[is greater than or equal to]2. The term “generalised Euler equations” is used to describe geodesic equations on Lie groups, which unifies many differential equations and has found many applications in such as hydrodynamics, medical imaging in the computational anatomy, and many other fields. The generalised Euler equations on the Bott-Virasoro group for k = 0, 1 are well-known and intensively studied— the Korteweg-de Vries equation for k = 0 and the Camassa-Holm equation for k = 1. Unlike these, the equations for k[is greater than or equal to]2, which we call the modified Camassa-Holm (mCH) equation, is not known to be integrable. This distinction motivates the study of the mCH equation. In this thesis, we derive the mCH equation and establish the short time existence of solutions, the well-posedness of the mCH equation, long time existence, the existence of the weak solutions, both on the circle S and [blackboard bold] R, and three conservation laws, show some quite interesting properties, for example, they do not lead to the blowup in finite time, unlike the Camassa-Holm equation. We then consider two numerical methods for the modified Camassa-Holm equation: the particle method and the box scheme. We prove the convergence result of the particle method. The numerical simulations indicate another interesting phenomenon: although mCH does not admit blowup in finite time, it admits solutions that blow up (which means their maximum value becomes infinity) at infinite time, which we call weak blowup. We study this novel phenomenon using the method of matched asymptotic expansion. A whole family of self-consistent blowup profiles is obtained. We propose a mechanism by which the actual profile is selected that is consistent with the simulations, but the mechanism is only partly supported by the analysis. We study the four particle systems for the mCH equation finding numerical evidence both for the non-integrability of the mCH equations and for the existence of the fourth integral. We also study the higher dimensional case and obtain the short time existence and well-posedness for the generalised Euler equation in the two dimension case.