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    An equation-free approach for heterogeneous networks : this dissertation is submitted for the degree of Doctor of Philosophy, School of Mathematical and Computational Sciences, Massey University
    (Massey University, 2023) Zafar, Sidra
    Oscillators exist in many biological, chemical and physical systems. Often when oscillators with different periods of oscillations are coupled, they synchronize and oscillate with the same period. Examples include groups of synchronously flashing fireflies or chirping crickets. There are two questions of interest in this work. (1) Under what conditions will a system of coupled oscillators synchronize? and (2) Can a large system of synchronized oscillators be represented by a smaller number of variables? We study these questions for Kuramoto like models which are coupled in different ways. Examples include spatially extended and all-all coupling. We study conditions under which synchronization occurs in small and large networks by varying the coupling strength, calculating stabilities of synchronized solutions and creating bifurcation diagrams of the steady state solution as a function of coupling strength in one and two-dimensions. We use an equation free approach to approximate the coarse scale behavior of a large, coupled network, for which the equations are known, by a low dimensional description of variables, for which no governing equations are available in closed form. Our results show that a small number of variables can reproduce the behavior of the stable solutions in the full systems. However, the equation-free approach did not work as well for the unstable solutions. Possible reasons for this are explored in the thesis.
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    Monotone iterative methods for solving nonlinear partial differential 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, 2020) Al-Sultani, Mohamed
    A key aspect of the simulation process is the formulation of proper mathematical models. The model must be able to emulate the physical phenomena under investigation. Partial differential equations play a major role in the modelling of many processes which arise in physics, chemistry and engineering. Most of these partial differential equations cannot be solved analytically and classical numerical methods are not always applicable. Thus, efficient and stable numerical approaches are needed. A fruitful method for solving the nonlinear difference schemes, which discretize the continuous problems, is the method of upper and lower solutions and its associated monotone iterations. By using upper and lower solutions as two initial iterations, one can construct two monotone sequences which converge monotonically from above and below to a solution of the problem. This monotone property ensures the theorem on existence and uniqueness of a solution. This method can be applied to a wide number of applied problems such as the enzyme-substrate reaction diffusion models, the chemical reactor models, the logistic model, the reactor dynamics of gasses, the Volterra-Lotka competition models in ecology and the Belousov-Zhabotinskii reaction diffusion models. In this thesis, for solving coupled systems of elliptic and parabolic equations with quasi-monotone reaction functions, we construct and investigate block monotone iterative methods incorporated with Jacobi and Gauss--Seidel methods, based on the method of upper and lower solutions. The idea of these methods is the decomposition technique which reduces a computational domain into a series of nonoverlapping one dimensional intervals by slicing the domain into a finite number of thin strips, and then solving a two-point boundary-value problem for each strip by a standard computational method such as the Thomas algorithm. We construct block monotone Jacobi and Gauss-Seidel iterative methods with quasi-monotone reaction functions and investigate their monotone properties. We prove theorems on existence and uniqueness of a solution, based on the monotone properties of iterative sequences. Comparison theorems on the rate of convergence for the block Jacobi and Gauss-Seidel methods are presented. We prove that the numerical solutions converge to the unique solutions of the corresponding continuous problems. We estimate the errors between the numerical and exact solutions of the nonlinear difference schemes, and the errors between the numerical solutions and the exact solutions of the corresponding continuous problems. The methods of construction of initial upper and lower solutions to start the block monotone iterative methods are given.
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    Convergence properties of Fock-space based approaches in strongly correlated Fermi gases : a dissertation presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand
    (Massey University, 2019) Jeszenszki, Peter
    The main objective of this thesis is the effcient numerical description of strongly correlated quantum gases. Due to the complex many-body structure of the wave function, usually, numerical methods are required for its computation. The exact diagonalization approach is considered, where the energies and the wave functions are obtained by diagonalizing the Hamiltonian in a many-body basis. The dimension of the space increases combinatorially with the number of particles and the number of single-particle basis functions, which limits the characterization of fewbody systems to intermediate interactions. One of the main components of the convergence rate originates from the particle-particle interaction itself. The bare contact interaction introduces a singularity in the wave function at the particleparticle coalescence point. This is responsible for the slow convergence in the nite basis expansion in one dimension and it even causes pathological behavior in higher dimensions. Firstly, the Gaussian interaction potential is examined as an alternative pseudopotential. After the description of the accurate calculation of the s-wave scattering length of this potential, the convergence properties are investigated. As this function is smooth, by construction the wave function is free from any singularity implying an exponentially fast convergence rate. If the resolution of the basis set is not fine enough, the finite-range pseudopotential is indistinguishable from the pathological contact potential. Through the example of few particles in a two-dimensional harmonic trap, we show that in order to reach the necessary resolution, the number of harmonic-oscillator single-particle basis functions must increase quadratically with the inverse characteristic length of the pseudopotential. This scaling property combined with the combinatorial growth of the many-body space makes the physically realistic short-range potentials computationally inaccessible. We have also applied the so-called transcorrelated approach, where the singular part of the wave function is isolated in a Jastrow-type factor. This factor can be transformed into the Hamiltonian reducing the irregularity of the eigenfunction and improving the convergence rate. We will show through the example of the homogeneous gas in one dimension that this transformation efficiently improves the convergence from M⁻¹ to M⁻³, where M is the number of the single-particle plane-wave basis functions.