This dissertation includes two separate topics. In the first part, we extend fast wavelets collocation method and multilevel augmentation method on three dimensional Hammerstein equation with both smooth kernel and weakly singular kernel. In this part, self similar partition on d-dimensional (d ≥ 3) unit cube will be introduced and followed by a group of three dimensional contractive mappings on unit cube and unit prism. Hence, three dimensional wavelets and collocation polynomial on unit cube and unit prism are constructed respectively. Theoretical truncation strategy and practical block truncation strategy are compared with respect to numerical error, convergence rate, compression ratio and computing time by several different groups of parameters.
In the second part of this dissertation, we propose fast degenerate kernel by combining the practical truncation strategy in [25] with degenerate kernel method developed in [23]. Legendre piecewise orthogonal wavelets have been used to approximate the kernel which leads sparse structure in the linear system of the Fredholm equation and Jacobian matrix in Hammerstein equation. A fast degenerate kernel method takes place once the practical block truncation strategy implemented. Numerical examples are given throughout this dissertation.
","abstract_html":"<p>This dissertation includes two separate topics. In the first part, we extend fast wavelets collocation method and multilevel augmentation method on three dimensional Hammerstein equation with both smooth kernel and weakly singular kernel. In this part, self similar partition on d-dimensional (<em>d</em> ≥ 3) unit cube will be introduced and followed by a group of three dimensional contractive mappings on unit cube and unit prism. Hence, three dimensional wavelets and collocation polynomial on unit cube and unit prism are constructed respectively. Theoretical truncation strategy and practical block truncation strategy are compared with respect to numerical error, convergence rate, compression ratio and computing time by several different groups of parameters.</p> <p>In the second part of this dissertation, we propose fast degenerate kernel by combining the practical truncation strategy in [25] with degenerate kernel method developed in [23]. Legendre piecewise orthogonal wavelets have been used to approximate the kernel which leads sparse structure in the linear system of the Fredholm equation and Jacobian matrix in Hammerstein equation. A fast degenerate kernel method takes place once the practical block truncation strategy implemented. Numerical examples are given throughout this dissertation.</p>","abstract_has_math":false,"creators":["Chen, Xingwang"],"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","Fang Q. Hu","Shizhi Qian"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-04-01T07:00:00Z","date_published":"2015-04-01T07:00:00Z","updated_at":"2026-07-24T03:35:30Z","subjects":["Degenerate kernel method","Hammerstein equation","Three-dimension","Wavelet collocation method","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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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["9781339126500","https://digitalcommons.odu.edu/mathstat_etds/111"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This dissertation includes two separate topics. In the first part, we extend fast wavelets collocation method and multilevel augmentation method on three dimensional Hammerstein equation with both smooth kernel and weakly singular kernel. In this part, self similar partition on d-dimensional (d ≥ 3) unit cube will be introduced and followed by a group of three dimensional contractive mappings on unit cube and unit prism. Hence, three dimensional wavelets and collocation polynomial on unit cube and unit prism are constructed respectively. Theoretical truncation strategy and practical block truncation strategy are compared with respect to numerical error, convergence rate, compression ratio and computing time by several different groups of parameters.
In the second part of this dissertation, we propose fast degenerate kernel by combining the practical truncation strategy in [25] with degenerate kernel method developed in [23]. Legendre piecewise orthogonal wavelets have been used to approximate the kernel which leads sparse structure in the linear system of the Fredholm equation and Jacobian matrix in Hammerstein equation. A fast degenerate kernel method takes place once the practical block truncation strategy implemented. Numerical examples are given throughout this dissertation.
"]},{"key":"dc:title","label":"Title","values":["Wavelet Collocation Method for Hammerstein Integral Equations of High Dimension"]}]}],"canonical_facts":{"dc:contributor":["Hideaki Kaneko","Richard D. Noren","Fang Q. Hu","Shizhi Qian"],"dc:creator":["Chen, Xingwang"],"dc:date.available":["2100-01-01T08:00:00Z"],"dc:description.abstract":["This dissertation includes two separate topics. In the first part, we extend fast wavelets collocation method and multilevel augmentation method on three dimensional Hammerstein equation with both smooth kernel and weakly singular kernel. In this part, self similar partition on d-dimensional (d ≥ 3) unit cube will be introduced and followed by a group of three dimensional contractive mappings on unit cube and unit prism. Hence, three dimensional wavelets and collocation polynomial on unit cube and unit prism are constructed respectively. Theoretical truncation strategy and practical block truncation strategy are compared with respect to numerical error, convergence rate, compression ratio and computing time by several different groups of parameters.
In the second part of this dissertation, we propose fast degenerate kernel by combining the practical truncation strategy in [25] with degenerate kernel method developed in [23]. Legendre piecewise orthogonal wavelets have been used to approximate the kernel which leads sparse structure in the linear system of the Fredholm equation and Jacobian matrix in Hammerstein equation. A fast degenerate kernel method takes place once the practical block truncation strategy implemented. Numerical examples are given throughout this dissertation.
"],"dc:identifier":["9781339126500","https://digitalcommons.odu.edu/mathstat_etds/111"],"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":["Degenerate kernel method","Hammerstein equation","Three-dimension","Wavelet collocation method","Mathematics"],"dc:title":["Wavelet Collocation Method for Hammerstein Integral Equations of High Dimension"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:35:30Z"}