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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.

Let Ω be an arbitrary open set in IR³. Let || • || denote the L²(Ω) norm, and let [formula omitted] denote the completion of [formula omitted] in the Dirichlet norm || ∇•||. The pointwise bound [forumula omitted] is established for all functions [formula omitted] with Δ u є L² (Ω). The constant [formula omitted] is shown to be the best possible. Previously, inequalities of this type were proven only for bounded smooth domains or convex domains, with constants depending on the regularity of the boundary. A new method is employed to obtain this sharp inequality. The key idea is to estimate the maximum value of the quotient ⃒u(x)⃒/ || ∇u || ½ || Δ u || ½, where the point x is fixed, and the function u varies in the span of a finite number of eigenfunctions of the Laplacian. This method admits generalizations to other elliptic operators and other domains. The inequality is applied to study the initial-boundary value problem for Burgers' equation: [formula omitted] in arbitrary domains, with initial data in [formula omitted]. New a priori estimates are obtained. Adapting and refining known theory for Navier-Stokes equations, the existence and uniqueness of bounded smooth solutions are established. As corollaries of the inequality and its proof, pointwise bounds are given for eigenfunctions of the Laplacian in terms of the corresponding eigenvalues in two- and three-dimensional domains. / Science, Faculty of / Mathematics, Department of / Graduate

The mathematical equations describing transient heat transfer between the fluid flowing through a fixed bed and a moving bed of packing were formulated. The resistance to heat transfer within the packing due to its finite thermal conductivity was taken into account. An approximate integral method was applied to obtain an analytical solution to transient response of the bed packing. Results for two cases of fixed and moving bed were obtained. The validity of the approximate method was checked against the more exact method employed by Handley and Heggs who obtained the results for a fixed bed of packing with a step change in fluid inlet temperature. It was concluded that the approximate method gives results that agree well with the more exact methods. The method considered here provides a quick determination of the packing mean temperature in order to obtain the effectiveness. The other peculiarity of this method is that the effect of packing thermal conductivity can be examined very quickly since the solution is in analytical form. The analysis of the results revealed that as the thermal conductivity of the packing decreases the difference between its surface and mean temperature increases. A series of charts showing the comparison between the packing surface and mean temperatures for different thermal conductivities are presented. The approximate method was a moving bed of packing. It was packing thermal conductivity is series of charts representing versus dimensionless length conductivities are presented. then applied to the case of concluded that the effect of more severe than expected. A the moving bed effectiveness for different thermal / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

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This paper is devoted to finding integral solutions of algebraic equations. Only algebraic equations with integral coefficients are considered. The elementary properties of integers are assumed.

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