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Iterative methods are considered for the numerical solution of large, sparse, nonsingular, and nonsymmetric systems of linear equations Ax=b, where it is also required that A is p-cyclic (p≥2). Firstly, it is shown that the SOR method applied to the system with A as p-cyclic, if p > 2, has a slower rate of convergence than the SOR method applied to the same system with A considered as 2-cyclic under some conditions. Therefore, the p-cyclic matrix A should be partitioned into 2-cyclic form when the SOR method is applied.

The thesis commences with a description and classification of partial differential equations and the related matrix and eigenvalue theory. In most all cases the study of parabolic equations leads to initial boundary value problems and it is to this problem that the thesis is mainly concerned with. The basic (finite difference) methods to solve a (parabolic) partial differential equation are presented in the second chapter which is then followed by particular types of parabolic equations such as diffusion-convection, fourth order and non-linear problems in the third chapter. An introduction to the finite element technique is also included as an alternative to the finite difference method of solution. The advantages and disadvantages of some different strategies in terms of stability and truncation error are also considered. In Chapter Four the general derivation of a two time-level finite difference approximation to the simple heat conduction equation is derived. A new class of methods called the Group Explicit (GE) method is established which improves the stability of the previous explicit method. Comparison between the two methods in this class and the previous methods is also given. The method is also used 1n solving the two-space dimensional parabolic equation. The derivation of a general two-time level finite difference approximation and the general idea of the Group Explicit method are extended to the diffusion-convection equation in Chapter Five. Some other explicit algorithms for solving this problem ar~ also considered. In the sixth chapter the Group Explicit procedure is applied to solve a fourth-order parabolic equation on two interlocking nets. The concept of the GE method is also extendable to a non-linear partial differential equation. Consideration of this extension to a particular problem can be found in Chapter Seven. In Chapter Eight, some work on the finite element method for solving the heat-conduction and diffusion-convection equation is presented. Comparison of the results from this method with the finite-difference methods is given. The formulation and solution of this problem as a boundary value problem by the boundary value technique is also considered. A special method for solving diffusion-convection equation is presented in Chapter Nine as well as an extension of the Group Explicit method to a hyperbolic partial differential equation is given. The thesis concludes with recommendations for further work.

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Algebra I encompasses several topics that serve as a basis for students’ subsequent mathematics courses as they progress in school. Some of the key topics that students struggle with is solving linear equations and algebraic word problems. There are several factors that may contribute to this ongoing struggle for students such as the structure of the textbooks, the teacher instruction and misconceptions of components of algebraic equations. A promising solution to the potential contributing factors is concrete fading. In this study, concreteness fading refers to an instructional technique that represents topics in a particular sequence from a concrete, real-world representation to a semi-concrete diagram (e.g., tape diagram) to an abstract representation (e.g., algebraic equations). The current study aims to investigate the influence concrete fading has on student learning while studying concrete fading in two ninth grade Algebra I general education classes at an urban high school. In particular, the study aims to answer the following: 1) What are some ways that students who received concrete fading think differently than the control group? 2) How do these differences seem to be related to the intervention? Both classes were taught by the same teacher. One class was assigned to the treatment group that received the concrete fading lessons and the other class was assigned to the control group that was taught as business as usual by the teacher. The study was intended to be quasi-experimental study, but due to challenges, it was primarily qualitative in nature focusing on eight students where the analysis included analyzing student work and student interviews responses along with quantitative analysis of the pre and two post-tests. Results revealed that the treatment group does think differently than the control group based on student work and the interview responses. / Math & Science Education