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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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    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.