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
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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
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