In this thesis numerical integration in one and two dimensions is
considered. In chapter two transformation methods are considered
primarily for singular integrals and methods of computing the
transformations themselves are derived. The well-known transformation
based on the IMT rule and error function are extended to non-standard
functions. The implementation of these rules and their performances
are demonstrated.
These transformations are then extended to two-dimensions and are
used to develop accurate rules for integrating singular integrals. In
addition to this, a polynomial transformation with the aim of the
reduction in the number of function evaluations is also considered
and the resultant product rule is applied to two-dimensional non-singular
integrals.
Finally, the use of monomials in the construction of integration
rules for non-singular two-dimensional integrals is considered and some
rules developed. In all these situations the rules developed are
tested and compared with existing methods. The results show that the
new rules compare favourably with existing ones
