The study of some numerical methods for solving partial differential equations

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.