Date of Award

Summer 1998

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

Program/Concentration

Computational and Applied Mathematics

Committee Director

Hideaki Kaneko

Committee Member

John J. Swettis

Committee Member

Richard D. Noren

Committee Member

Fang Q. Hu

Committee Member

Doug Price

Abstract

In this thesis, we investigate the superconvergence phenomenon of the iterated numerical solutions for the Fredholm integral equations of the second kind as well as a class of nonliner Hammerstein equations. The term superconvergence was first described in the early 70s in connection with the solution of two-point boundary value problems and other related partial differential equations. Superconvergence in this context was understood to mean that the order of convergence of the numerical solutions arising from the Galerkin as well as the collocation method is higher at the knots than we might expect from the numerical solutions that are obtained by applying a class of piecewise polynomials as approximating functions. The type of superconvergence that we investigate in this thesis is different. We are interested in finding out whether or not we obtain an enhancement in the global rate of convergence when the numerical solutions are iterated through integral operators. A general operator approximation scheme for the second kind linear equation is described that can be used to explain some of the existing superconvergence results. Moreover, a corollary to the general approximation scheme will be given which can be used to establish the superconvergence of the iterated degenerate kernel method for the Fredholm equations of the second kind. We review the iterated Galerkin method for Hammerstein equations and discuss the iterated degenerate kernel method for the Fredholm equations of the second kind. We review the iterated Galerkin method for Hammerstein equations and discuss the iterated degenerate kernel method for Hammerstein and weakly singular Hammerstein equations and its corresponding superconvergence phenomena for the iterated solutions. The type of regularities that the solution of weakly singular Hammerstein equations possess is investigated. Subsequently, we establish the singularity preserving Galerkin method for Hammerstein equations. Finally, the superconvergence results for the iterated solutions corresponding to this method will be described.

DOI

10.25777/f74r-nf52

ISBN

9780599059580

Included in

Mathematics Commons

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