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
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.
A Doctoral Thesis. Submitted in partial fulfillment of the requirements for the award of Doctor of Philosophy of Loughborough University.