Weighted mean iterative methods for solving Fredholm integral equations

Integral equations (IEs) are used as mathematical models for many and varied physical circumstances, and also occur as reformulations of other mathematicalproblems. In this research, first and second kind linear IEs of Fredholm type are considered and solved using numerical approaches. The essential...

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Main Author:	Mohana Sundaram Muthuvalu
Format:	Thesis
Language:	English
Published:	2012
Subjects:	QA Mathematics
Online Access:	https://eprints.ums.edu.my/id/eprint/11798/1/ph000000076.pdf
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spelling	my-ums-ep.117982017-11-07T07:07:58Z Weighted mean iterative methods for solving Fredholm integral equations 2012 Mohana Sundaram Muthuvalu QA Mathematics Integral equations (IEs) are used as mathematical models for many and varied physical circumstances, and also occur as reformulations of other mathematicalproblems. In this research, first and second kind linear IEs of Fredholm type are considered and solved using numerical approaches. The essential aim of this research was to investigate the effectiveness of the point and block Weighted Mean (WM) iterative methods categorized as two-stage iterative methods in solving linear systems generated from the discretization of the first and second kind linear Fredholm integral equations (FIEs). In the aspect of discretization schemes, three schemes of different order under composite closed Newton-Cotes quadrature and piecewise polynomial collocation methods were used to discretize first and second kind linear FIEs. Moreover, discussions on computational complexity of the tested point and block WM methods in this research were also included. By comparing point WM iterative methods, the point methods under Geometric Mean (GM) and Harmonic Mean (HM) families are slightly superior to equivalent Arithmetic Mean (AM) methods, particularly for first kind linear FIEs. Meanwhile, performance of the point GM and HM methods is comparable. Based on numerical experiments, results show that proposed 6-Point Quarter-Sweep Block Arithmetic Mean (6-QSBLAM), 6-Point Quarter-Sweep Block Geometric Mean (6-QSBLGM) and 6-Point Quarter Sweep Block Harmonic Mean (6-QSBLHM) methods are the best tested AM, GM and HM iterative methods respectively in solving composite closed Newton-Cotes quadrature and piecewise polynomial collocation systems associated with numerical solutions of first and second kind linear FIEs in the sense of number of iterations and CPU time. For comparison purpose among 6-Point Quarter-Sweep Block Weighted Mean (6-QSBLWM) methods, 6-QSBLGM and 6-QSBLHM methods are slightly better than 6-QSBLAM method in solving FIEs. All variants of point and block WM methods, which were formulated using the half- and quarter-sweep iteration concepts reduce the computational complexity of the standard WM iterative methods at least 75% and 93.75% respectively. in terms of accuracy, all three schemes under piecewise polynomial collocation method yields more accurate approximation solutions than composite closed Newton-Cotes quadrature schemes particularly for the first kind FIEs problems. However, by comparing corresponding orders of composite closed Newton-Cotes quadrature and piecewise polynomial collocation schemes, the accuracy of the approximation solutions is comparable when solving second kind linear FIEs. 2012 Thesis https://eprints.ums.edu.my/id/eprint/11798/ https://eprints.ums.edu.my/id/eprint/11798/1/ph000000076.pdf text en public phd doctoral Universiti Malaysia Sabah School of Science and Technology
institution	Universiti Malaysia Sabah
collection	UMS Institutional Repository
language	English
topic	QA Mathematics
spellingShingle	QA Mathematics Mohana Sundaram Muthuvalu Weighted mean iterative methods for solving Fredholm integral equations
description	Integral equations (IEs) are used as mathematical models for many and varied physical circumstances, and also occur as reformulations of other mathematicalproblems. In this research, first and second kind linear IEs of Fredholm type are considered and solved using numerical approaches. The essential aim of this research was to investigate the effectiveness of the point and block Weighted Mean (WM) iterative methods categorized as two-stage iterative methods in solving linear systems generated from the discretization of the first and second kind linear Fredholm integral equations (FIEs). In the aspect of discretization schemes, three schemes of different order under composite closed Newton-Cotes quadrature and piecewise polynomial collocation methods were used to discretize first and second kind linear FIEs. Moreover, discussions on computational complexity of the tested point and block WM methods in this research were also included. By comparing point WM iterative methods, the point methods under Geometric Mean (GM) and Harmonic Mean (HM) families are slightly superior to equivalent Arithmetic Mean (AM) methods, particularly for first kind linear FIEs. Meanwhile, performance of the point GM and HM methods is comparable. Based on numerical experiments, results show that proposed 6-Point Quarter-Sweep Block Arithmetic Mean (6-QSBLAM), 6-Point Quarter-Sweep Block Geometric Mean (6-QSBLGM) and 6-Point Quarter Sweep Block Harmonic Mean (6-QSBLHM) methods are the best tested AM, GM and HM iterative methods respectively in solving composite closed Newton-Cotes quadrature and piecewise polynomial collocation systems associated with numerical solutions of first and second kind linear FIEs in the sense of number of iterations and CPU time. For comparison purpose among 6-Point Quarter-Sweep Block Weighted Mean (6-QSBLWM) methods, 6-QSBLGM and 6-QSBLHM methods are slightly better than 6-QSBLAM method in solving FIEs. All variants of point and block WM methods, which were formulated using the half- and quarter-sweep iteration concepts reduce the computational complexity of the standard WM iterative methods at least 75% and 93.75% respectively. in terms of accuracy, all three schemes under piecewise polynomial collocation method yields more accurate approximation solutions than composite closed Newton-Cotes quadrature schemes particularly for the first kind FIEs problems. However, by comparing corresponding orders of composite closed Newton-Cotes quadrature and piecewise polynomial collocation schemes, the accuracy of the approximation solutions is comparable when solving second kind linear FIEs.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Mohana Sundaram Muthuvalu
author_facet	Mohana Sundaram Muthuvalu
author_sort	Mohana Sundaram Muthuvalu
title	Weighted mean iterative methods for solving Fredholm integral equations
title_short	Weighted mean iterative methods for solving Fredholm integral equations
title_full	Weighted mean iterative methods for solving Fredholm integral equations
title_fullStr	Weighted mean iterative methods for solving Fredholm integral equations
title_full_unstemmed	Weighted mean iterative methods for solving Fredholm integral equations
title_sort	weighted mean iterative methods for solving fredholm integral equations
granting_institution	Universiti Malaysia Sabah
granting_department	School of Science and Technology
publishDate	2012
url	https://eprints.ums.edu.my/id/eprint/11798/1/ph000000076.pdf
_version_	1747836402340462592

Weighted mean iterative methods for solving Fredholm integral equations

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