In this dissertation, we develop the Petrov-Galerkin method and the iterated Petrov-Galerkin method for a class of nonlinear Hammerstein equation. We also investigate the superconvergence phenomenon of the iterated Petrov-Galerkin and degenerate kernel numerical solutions of linear and nonlinear integral equations with a class of wavelet basis. The Fredholm integral equations and the Hammerstein equations are considered in linear and nonlinear cases respectively. Alpert demonstrated that an application of a class of wavelet basis elements in the Galerkin approximation of the Fredholm equation of the second kind leads to a system of linear equations which is sparse. The main concern of this dissertation is to address the issue of how this sparsity manifests itself in the setting of nonlinear equations, particularly for Hammerstein equations. We demonstrate that sparsity appears in the Jacobian matrix when one attempts to solve the system of nonlinear equations by the Newton's iterative method. Overall, the dissertation generalizes the results of Alpert to nonlinear equations setting as well as the results of Chen and Xu, who discussed the Petrov-Galerkin method for Fredholm equation, to nonlinear equations setting.
","abstract_html":"<p>In this dissertation, we develop the Petrov-Galerkin method and the iterated Petrov-Galerkin method for a class of nonlinear Hammerstein equation. We also investigate the superconvergence phenomenon of the iterated Petrov-Galerkin and degenerate kernel numerical solutions of linear and nonlinear integral equations with a class of wavelet basis. The Fredholm integral equations and the Hammerstein equations are considered in linear and nonlinear cases respectively. Alpert demonstrated that an application of a class of wavelet basis elements in the Galerkin approximation of the Fredholm equation of the second kind leads to a system of linear equations which is sparse. The main concern of this dissertation is to address the issue of how this sparsity manifests itself in the setting of nonlinear equations, particularly for Hammerstein equations. We demonstrate that sparsity appears in the Jacobian matrix when one attempts to solve the system of nonlinear equations by the Newton's iterative method. Overall, the dissertation generalizes the results of Alpert to nonlinear equations setting as well as the results of Chen and Xu, who discussed the Petrov-Galerkin method for Fredholm equation, to nonlinear equations setting.</p>","abstract_has_math":false,"creators":["Novaprateep, Boriboon"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":["Hideaki Kaneko","Richard D. Noren","Charlie H. Cooke","Fang Q. Hu","Duc T. Nguyen"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2003,"date_issued":"2003-07-01T07:00:00Z","date_published":"2003-07-01T07:00:00Z","updated_at":"2026-07-24T03:34:53Z","subjects":["Integral equations","Iterated solutions","Linear integral equations","Nonlinear integral equations","Superconvergence","Wavelet","Mathematics"],"languages":[],"rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
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"]},{"key":"dc:title","label":"Title","values":["Superconvergence of Iterated Solutions for Linear and Nonlinear Integral Equations: Wavelet Applications"]}]}],"canonical_facts":{"dc:contributor":["Hideaki Kaneko","Richard D. Noren","Charlie H. Cooke","Fang Q. Hu","Duc T. Nguyen"],"dc:creator":["Novaprateep, Boriboon"],"dc:date.available":["2019-06-07T07:00:00Z"],"dc:description.abstract":["In this dissertation, we develop the Petrov-Galerkin method and the iterated Petrov-Galerkin method for a class of nonlinear Hammerstein equation. We also investigate the superconvergence phenomenon of the iterated Petrov-Galerkin and degenerate kernel numerical solutions of linear and nonlinear integral equations with a class of wavelet basis. The Fredholm integral equations and the Hammerstein equations are considered in linear and nonlinear cases respectively. Alpert demonstrated that an application of a class of wavelet basis elements in the Galerkin approximation of the Fredholm equation of the second kind leads to a system of linear equations which is sparse. The main concern of this dissertation is to address the issue of how this sparsity manifests itself in the setting of nonlinear equations, particularly for Hammerstein equations. We demonstrate that sparsity appears in the Jacobian matrix when one attempts to solve the system of nonlinear equations by the Newton's iterative method. Overall, the dissertation generalizes the results of Alpert to nonlinear equations setting as well as the results of Chen and Xu, who discussed the Petrov-Galerkin method for Fredholm equation, to nonlinear equations setting.
"],"dc:identifier":["9780496584963","https://digitalcommons.odu.edu/mathstat_etds/31"],"dc:rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
"],"dc:subject":["Integral equations","Iterated solutions","Linear integral equations","Nonlinear integral equations","Superconvergence","Wavelet","Mathematics"],"dc:title":["Superconvergence of Iterated Solutions for Linear and Nonlinear Integral Equations: Wavelet Applications"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:34:53Z"}