Kress smoothing transformation for weakly singular Fredholm integral equation of second kind
This work investigates a numerical method for the second kind Fredholm integral equation with weakly singular kernel k(x, y), in particular, when k(x, y) = ln |x-y|, and k(x, y) = |x-y|-a, -1 = x, y = 1, 0 < a < 1. The solutions of such equations may exhibit a singular behaviour in the neighbo...
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| Format: | Thesis |
| Language: | English |
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2006
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| Online Access: | http://eprints.utm.my/5302/ http://eprints.utm.my/5302/10/HassanMohamedSaeedBawazirMFS2006.pdf |
| _version_ | 1848891017135128576 |
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| author | Mohamed Saeed Bawazir, Hassan |
| author_facet | Mohamed Saeed Bawazir, Hassan |
| author_sort | Mohamed Saeed Bawazir, Hassan |
| building | UTeM Institutional Repository |
| collection | Online Access |
| description | This work investigates a numerical method for the second kind Fredholm integral equation with weakly singular kernel k(x, y), in particular, when k(x, y) = ln |x-y|, and k(x, y) = |x-y|-a, -1 = x, y = 1, 0 < a < 1. The solutions of such equations may exhibit a singular behaviour in the neighbourhood of the endpoints x = ±1. We introduce a new smoothing transformation based on the Kress transformation for solving weakly singular Fredholm integral equations of the second kind, and then using the Hermite smoothing transformation as a standard. With the transformation an equation which is still weakly singular is obtained, but whose solution is smoother. The transformed equation is then solved numerically by product integration methods with interpolating polynomials. Two types of interpolating polynomials, namely the Gauss-Legendre and Chebyshev polynomials, have been used. Numerical examples are presented to investigate the performance of the former. |
| first_indexed | 2025-11-15T20:51:16Z |
| format | Thesis |
| id | utm-5302 |
| institution | Universiti Teknologi Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T20:51:16Z |
| publishDate | 2006 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | utm-53022017-10-12T04:43:55Z http://eprints.utm.my/5302/ Kress smoothing transformation for weakly singular Fredholm integral equation of second kind Mohamed Saeed Bawazir, Hassan QA Mathematics This work investigates a numerical method for the second kind Fredholm integral equation with weakly singular kernel k(x, y), in particular, when k(x, y) = ln |x-y|, and k(x, y) = |x-y|-a, -1 = x, y = 1, 0 < a < 1. The solutions of such equations may exhibit a singular behaviour in the neighbourhood of the endpoints x = ±1. We introduce a new smoothing transformation based on the Kress transformation for solving weakly singular Fredholm integral equations of the second kind, and then using the Hermite smoothing transformation as a standard. With the transformation an equation which is still weakly singular is obtained, but whose solution is smoother. The transformed equation is then solved numerically by product integration methods with interpolating polynomials. Two types of interpolating polynomials, namely the Gauss-Legendre and Chebyshev polynomials, have been used. Numerical examples are presented to investigate the performance of the former. 2006-03 Thesis NonPeerReviewed application/pdf en http://eprints.utm.my/5302/10/HassanMohamedSaeedBawazirMFS2006.pdf Mohamed Saeed Bawazir, Hassan (2006) Kress smoothing transformation for weakly singular Fredholm integral equation of second kind. Masters thesis, Universiti Teknologi Malaysia, Faculty of Science. |
| spellingShingle | QA Mathematics Mohamed Saeed Bawazir, Hassan Kress smoothing transformation for weakly singular Fredholm integral equation of second kind |
| title | Kress smoothing transformation for weakly singular Fredholm integral equation of second kind |
| title_full | Kress smoothing transformation for weakly singular Fredholm integral equation of second kind |
| title_fullStr | Kress smoothing transformation for weakly singular Fredholm integral equation of second kind |
| title_full_unstemmed | Kress smoothing transformation for weakly singular Fredholm integral equation of second kind |
| title_short | Kress smoothing transformation for weakly singular Fredholm integral equation of second kind |
| title_sort | kress smoothing transformation for weakly singular fredholm integral equation of second kind |
| topic | QA Mathematics |
| url | http://eprints.utm.my/5302/ http://eprints.utm.my/5302/10/HassanMohamedSaeedBawazirMFS2006.pdf |