Wavelet Collocation Method for Hammerstein Integral Equations of High Dimension

Date of Award

Spring 2015

Document Type


Degree Name

Doctor of Philosophy (PhD)


Mathematics and Statistics


Computational and Applied Mathematics

Committee Director

Hideaki Kaneko

Committee Member

Richard D. Noren

Committee Member

Fang Q. Hu

Committee Member

Shizhi Qian


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.





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