### Abstract:

Spectral integration methods have been introduced for constant-coefficient
two-point boundary value problems by Greengard, and pseudospectral integration
methods for Volterra integral equations have been investigated by Kauthen. This thesis
presents an approach to variable-coefficient two-point boundary value problems which
employs pseudospectral integration methods to solve an equivalent integral equation.
This thesis covers three topics in the application of spectral integration methods to
two-point boundary value problems.
The first topic is the development of the spectral integration concept and a
derivation of the spectral integration matrices. The derivation utilizes the discrete
Chebyshev transform and leads to a stable algorithm for generating the integration
matrices. Convergence theory for spectral integration of C[subscript k] and analytic functions is
presented. Matrix-free implementations are discussed with an emphasis on
computational efficiency.
The second topic is the transformation of boundary value problems to equivalent
Fredholm integral equations and discretization of the resulting integral equations. The
discussion of boundary condition treatments includes Dirichlet, Neumann, and Robin
type boundary conditions.
The final topic is a numerical comparison of the spectral integration and spectral
differentiation approaches to two-point boundary value problems. Numerical results are
presented on the accuracy and efficiency of these two methods applied to a set of model
problems.
The main theoretical result of this thesis is a proof that the error in spectral
integration of analytic functions decays exponentially with the number of discretization
points N. It is demonstrated that spectrally accurate solutions to variable-coefficient
boundary value problems can be obtained in O(NlogN) operations by the spectral
integration method. Numerical examples show that spectral integration methods are
competitive with spectral differentiation methods in terms of accuracy and efficiency.