Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson's Rule
This paper discusses the use of Product Simpson’s rule to solve the integral equation eigenvalue problem f(x) = R 1−1 k(|x − y|)f(y)dy where k(t) = ln|t| or k(t) = t−, 0 < < 1, , f and are unknowns which we wish to obtain. The function f(y) in the integral above is replaced by an int...
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Department of Mathematics, Faculty of Science
2002
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| Online Access: | http://eprints.utm.my/2435/1/tryversion2007_NumericalExperimentsonEigenvalues.pdf http://eprints.utm.my/2435/ http://www.fs.utm.my/matematika/content/view/65/31/ |
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| author | Razali, Mohamad Rashidi Mohamed, M. S. Nasser |
| author_facet | Razali, Mohamad Rashidi Mohamed, M. S. Nasser |
| author_sort | Razali, Mohamad Rashidi |
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| content_provider | Universiti Teknologi Malaysia |
| content_source | UTM Institutional Repository |
| continent | Asia |
| country | Malaysia |
| description | This paper discusses the use of Product Simpson’s rule to solve the integral equation eigenvalue problem f(x) =
R 1−1 k(|x − y|)f(y)dy where k(t) = ln|t| or k(t) = t−, 0 < < 1, , f and are unknowns which we wish to obtain. The function f(y) in the integral above is replaced by an interpolating function Lfn(y) = Pni=0 f(xi)i(y), where i(y) are Simpson interpolating elements and x0, x1, . . . ,xn are the interpolating points and they are chosen to be the appropriate non-uniform mesh points in [−1, 1]. The product integration formula R 1−1 k(y)f(y)dy Pni=0 wif(xi) is used, where the weights wi are chosen such that the formula is exact when f(y) is replaced by Lf n(y) and k(y)as given above. The five eigenvalues with largest moduli of the two kernels K(x, y) = ln|x − y| and K(x, y) = |x − y|−, 0 < < 1 are given. Keywords eigenvalue, product integration, singular kernel, integral equation. |
| format | Article |
| id | my.utm.eprints-2435 |
| institution | Universiti Teknologi Malaysia |
| language | en |
| publishDate | 2002 |
| publisher | Department of Mathematics, Faculty of Science |
| record_format | eprints |
| spelling | my.utm.eprints-24352011-01-06T02:04:40Z http://eprints.utm.my/2435/ Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson's Rule Razali, Mohamad Rashidi Mohamed, M. S. Nasser QA Mathematics This paper discusses the use of Product Simpson’s rule to solve the integral equation eigenvalue problem f(x) = R 1−1 k(|x − y|)f(y)dy where k(t) = ln|t| or k(t) = t−, 0 < < 1, , f and are unknowns which we wish to obtain. The function f(y) in the integral above is replaced by an interpolating function Lfn(y) = Pni=0 f(xi)i(y), where i(y) are Simpson interpolating elements and x0, x1, . . . ,xn are the interpolating points and they are chosen to be the appropriate non-uniform mesh points in [−1, 1]. The product integration formula R 1−1 k(y)f(y)dy Pni=0 wif(xi) is used, where the weights wi are chosen such that the formula is exact when f(y) is replaced by Lf n(y) and k(y)as given above. The five eigenvalues with largest moduli of the two kernels K(x, y) = ln|x − y| and K(x, y) = |x − y|−, 0 < < 1 are given. Keywords eigenvalue, product integration, singular kernel, integral equation. Department of Mathematics, Faculty of Science 2002-06 Article PeerReviewed application/pdf en http://eprints.utm.my/2435/1/tryversion2007_NumericalExperimentsonEigenvalues.pdf Razali, Mohamad Rashidi and Mohamed, M. S. Nasser (2002) Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson's Rule. Matematika, 18 (1). pp. 9-20. ISSN 0127-8274 http://www.fs.utm.my/matematika/content/view/65/31/ |
| spellingShingle | QA Mathematics Razali, Mohamad Rashidi Mohamed, M. S. Nasser Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson's Rule |
| title | Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson's Rule |
| title_full | Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson's Rule |
| title_fullStr | Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson's Rule |
| title_full_unstemmed | Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson's Rule |
| title_short | Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson's Rule |
| title_sort | numerical experiments on eigenvalues of weakly singular integral equations using product simpson's rule |
| topic | QA Mathematics |
| url | http://eprints.utm.my/2435/1/tryversion2007_NumericalExperimentsonEigenvalues.pdf http://eprints.utm.my/2435/ http://www.fs.utm.my/matematika/content/view/65/31/ |
| url_provider | http://eprints.utm.my/ |