Recent surveys have revealed that the majority of numerical
methods for the solution of integral equations use one of two
main techniques for generating a set of simultaneous
equations for their solution. Either the unknown function is
expanded as a combination of basis set functions and the
resulting coefficients found, or the integral is discretized
using quadrature formulae. The latter results in simultaneous
equations for the solution at the quadrature abscissae.
The thesis proposes techniques based on various direct iterative
methods, including refinements of residual correction which
hold no restrictions for nonlinear integral equations. New
implementations of successive approximations and Newton's
method appear. The latter compares particularly well with other
versions as the evaluation of the Jacobian can be made
equivalent to the solution of matrix equations of relatively
small dimensions. The method can be adapted to the solution of
first-kind equations and has been applied to systems of integral
equations. The schemes are designed to be adaptive with the aid
of the progressive quadrature rules of Patterson or Clenshaw and
Curtis and interpolation formulae. The Clenshaw-Curtis rule is
particularly favoured as it delivers error estimates.
A very powerful routine for the solution of a wide range of integral equations has resulted with the inclusion of a new
efficient method for calculating singular integrals.
Some work is devoted to the conversion of differential to
integral or integro-differential equations and comparing the
merits of solving a problem in its original and converted forms.
Many equations are solved as test examples throughout the thesis
of which several are of physical significance. They include
integral equations for the slowing down of neutrons, the
Lane-Emden equation, an equation arising from a chemical reactor
problem, Chandrasekhar's isotropic scatter ing of radiation
equation and the Blasius equation in boundary layer theory.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.