The first chapter of the thesis is concerned with the construction of finite difference approximations to boundary value problems in linear nth order ordinary differential equations. This construction is based upon a local collocation procedure with polynomials, which is equivalent to a method of undetermined coefficients. It is shown that the coefficients of these finite difference approximations can be expressed as the determinants of matrices of relatively small dimension. A basic theorem states that these approximations are consistent, provided only that a certain normalization factor does not vanish. This is the case for compact difference equations and for difference equations with only one collocation point. The order of consistency may be improved by suitable choice of the collocation points. Several examples of known, as well as new difference approximations are given. Approximations to boundary conditions are also treated in detail. The stability theory of H. O. Kreiss is applied to investigate the stability of finite difference schemes based upon these approximations. A number of numerical examples are also given. In the second chapter it is shown how the construction method of the first chapter can be extended to initial value problems for systems of linear first order ordinary differential equations. Specific examples are 'included and the well-known stability theory for these difference equations is summarized. It is then shown how these difference methods may be applied to linear parabolic partial differential equations in one space variable after first discretizing in space by a suitable method from the first chapter. The stability of such difference schemes for parabolic equations is investigated using an eigenvalue-eigenvector analysis. In particular, the effect of various approximations to the boundary conditions is considered. The relation of this analysis to the stability theory of J. M. Varah is indicated. Numerical examples are also included. / Science, Faculty of / Mathematics, Department of / Graduate

The augmented phase portrait, introduced by Sabrina Streipert and Gail Wolkowicz, is used to analyze second order rational discrete maps of the form \begin{align*} x_{n+1} = \frac{\alpha + \beta x_n + \gamma x_{n-1}}{A + Bx_n + C x_{n-1}}, \text{ for } n \in \mathbb{N}_0 =\{0,1,2,\dots, \} \end{align*} with parameters $\alpha, \, \beta, \, \gamma, \, A, \, B, \, C \geq 0$, and initial conditions, $x_{0}, \, x_{-1} > 0$. First we study the special case, \begin{align*} x_{n+1} = \frac{\alpha + \gamma x_{n-1}}{A + Bx_n}, \end{align*} with $\alpha, \, \gamma, \, B > 0$ and $A \geq 0$. Applying the change of variables, $y_n = x_{n-1}$, this equation can be rewritten as a planar system. We provide a new proof to show that oscillatory solutions have semicycles of length one, except possibly the first cycle, and that nonoscillatory solutions must converge monotonically to the equilibrium. This was originally done by Gibbons, Kulenovic, and Ladas. We also show that when the unique positive equilibrium is a saddle point, there exist nontrivial positive solutions that increase and decrease monotonically to the equilibrium, proving Conjecture 5.4.6 from the monograph by Kulenovic and Ladas. In particular, Theorem 1.2 from this monograph defines the tangent vector to the stable manifold at the equilibrium. We show that specific regions defined by the augmented phase portrait have solutions that increase and decrease monotonically to the equilibrium along the stable manifold. While Conjecture 5.4.6 was previously proven in a paper by Hoag and a paper by Sun and Xi, our proof provides a more intuitive and elementary solution. We then consider the case, \begin{equation*} x_{n+1} = \frac{\alpha + \beta (x_n + x_{n-1})}{A + B(x_n + x_{n-1})}, \end{equation*} with $\alpha, \beta, A, B > 0$. Again, using $y_n = x_{n-1}$, this system can be written as a planar system. Thus, applying the augmented phase plane, we prove global asymptotic stability of the positive equilibrium for some cases. In other cases, we show this using other theorems from the monograph by Kulenovic and Ladas as was previously done by Atawna, et al. / Thesis / Master of Science (MSc) / The augmented phase portrait, introduced by Sabrina Streipert and Gail Wolkowicz, is used to analyze second order rational discrete maps with nonnegative parameters and positive initial conditions. Using a change of variable to transform the second order rational discrete maps into planar maps, various properties of solutions were analyzed for various cases. For one case, we provide a new proof to show that oscillatory solutions have semicycles of length one, except possibly the first cycle, and that nonoscillatory solutions must converge monotonically to the equilibrium. This was originally done by Gibbons, Kulenovic, and Ladas. We also prove Conjecture 5.4.6 from the monograph by Kulenovic and Ladas, showing the existence of solutions that increase and decrease monotonically to the equilibrium. While Conjecture 5.4.6 was previously proved in a paper by Hoag and a paper by Sun and Xi, our proof provides a more intuitive and elementary solution. Finally, for another case we prove global asymptotic stability of the positive equilibrium using the augmented phase portrait. Sometimes, we show this using other theorems from the monograph by Kulenovic and Ladas as was previously done by Atawna, et al.

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A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. September 26, 2014. / The aim of the dissertation is to extend on the work done by Hydon in [17]. We only consider second order ordinary difference equations and calculate their symmetry generators, first integrals and reduce their order, that is, find a general solution. We investigate the association between a symmetry generator and a first integral. Furthermore, we investigate when a reduced equation may be further reduced and lead to a double reduction. The examples considered are obtained from [17]. ii

Graduation date: 1996

Thesis (M.A..)--Marshall University, 2009. / Title from document title page. Includes abstract. Document formatted into pages: contains 95 pages. Includes bibliographical references p. 94-95.

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Non-negative matrices arise naturally in population models. In this thesis, we first study Perron- Frobenius theory of non-negative irreducible matrices. We use this theory to investigate the asymptotic behaviour of discrete time linear autonomous models. Then we discuss an application for this in age structured population. Furthermore, we study Liapunov stability of a general non-linear autonomous model. We consider a general nonlinear autonomous model that arises in structured population. We assume that the associated nonlinear matrix of this model is non-increasing at all density levels. Then, we show the existence of global extinction. In addition, we show the stability condition of the extinction equilibrium of the this model in the Liapunov sense. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2013.

Thesis (Ph.D.)--Baylor University, 2006. / In abstract "th, n, i, n-2, n-1" are superscript. Includes bibliographical references (p. 42-44).

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.