Development of numerical methods for the solution of integral equations
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.