Approximating Cauchy-type singular integral by an automatic quadrature scheme.
An automatic quadrature scheme is developed for the approximate evaluation of the product-type indefinite integral where View the MathML source, K(t,c)=1/(t−c) and f(t) is assumed to be a smooth function. In constructing an automatic quadrature scheme, we consider two cases: (1) −1<x<y<1, a...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English English |
| Published: |
Elsevier
2011
|
| Online Access: | http://psasir.upm.edu.my/id/eprint/24849/ http://psasir.upm.edu.my/id/eprint/24849/1/Approximating%20Cauchy.pdf |
| _version_ | 1848845148409036800 |
|---|---|
| author | Eshkuratov, Zainidin K. Ahmedov, Anvarjon Nik Long, Nik Mohd Asri Jamaludin, Nur Amalina |
| author_facet | Eshkuratov, Zainidin K. Ahmedov, Anvarjon Nik Long, Nik Mohd Asri Jamaludin, Nur Amalina |
| author_sort | Eshkuratov, Zainidin K. |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | An automatic quadrature scheme is developed for the approximate evaluation of the product-type indefinite integral where View the MathML source, K(t,c)=1/(t−c) and f(t) is assumed to be a smooth function. In constructing an automatic quadrature scheme, we consider two cases: (1) −1<x<y<1, and (2) x=−1,y=1. In both cases the density function f(t) is replaced by the truncated Chebyshev polynomial pN(t) of the first kind of degree N. The approximation pN(t) yields an integration rule QN(f,x,y,c) to the integral Q(f,x,y,c). Interpolation conditions are imposed to determine the unknown coefficients of the Chebyshev polynomials pN(t). Convergence problem of the approximate method is discussed in the classes of function CN+1,α[−1,1] and View the MathML source. Numerically, it is found that when the singular point c either lies in or outside the interval (x,y) or comes closer to the end points of the interval [−1,1], the proposed scheme gives a very good agreement with the exact solution. These results in the line of theoretical findings. |
| first_indexed | 2025-11-15T08:42:12Z |
| format | Article |
| id | upm-24849 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English English |
| last_indexed | 2025-11-15T08:42:12Z |
| publishDate | 2011 |
| publisher | Elsevier |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-248492015-09-09T04:57:45Z http://psasir.upm.edu.my/id/eprint/24849/ Approximating Cauchy-type singular integral by an automatic quadrature scheme. Eshkuratov, Zainidin K. Ahmedov, Anvarjon Nik Long, Nik Mohd Asri Jamaludin, Nur Amalina An automatic quadrature scheme is developed for the approximate evaluation of the product-type indefinite integral where View the MathML source, K(t,c)=1/(t−c) and f(t) is assumed to be a smooth function. In constructing an automatic quadrature scheme, we consider two cases: (1) −1<x<y<1, and (2) x=−1,y=1. In both cases the density function f(t) is replaced by the truncated Chebyshev polynomial pN(t) of the first kind of degree N. The approximation pN(t) yields an integration rule QN(f,x,y,c) to the integral Q(f,x,y,c). Interpolation conditions are imposed to determine the unknown coefficients of the Chebyshev polynomials pN(t). Convergence problem of the approximate method is discussed in the classes of function CN+1,α[−1,1] and View the MathML source. Numerically, it is found that when the singular point c either lies in or outside the interval (x,y) or comes closer to the end points of the interval [−1,1], the proposed scheme gives a very good agreement with the exact solution. These results in the line of theoretical findings. Elsevier 2011 Article PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/24849/1/Approximating%20Cauchy.pdf Eshkuratov, Zainidin K. and Ahmedov, Anvarjon and Nik Long, Nik Mohd Asri and Jamaludin, Nur Amalina (2011) Approximating Cauchy-type singular integral by an automatic quadrature scheme. Journal of Computational and Applied Mathematics, 235 (16). pp. 4675-4686. ISSN 0377-0427 10.1016/j.cam.2010.08.001 English |
| spellingShingle | Eshkuratov, Zainidin K. Ahmedov, Anvarjon Nik Long, Nik Mohd Asri Jamaludin, Nur Amalina Approximating Cauchy-type singular integral by an automatic quadrature scheme. |
| title | Approximating Cauchy-type singular integral by an automatic quadrature scheme. |
| title_full | Approximating Cauchy-type singular integral by an automatic quadrature scheme. |
| title_fullStr | Approximating Cauchy-type singular integral by an automatic quadrature scheme. |
| title_full_unstemmed | Approximating Cauchy-type singular integral by an automatic quadrature scheme. |
| title_short | Approximating Cauchy-type singular integral by an automatic quadrature scheme. |
| title_sort | approximating cauchy-type singular integral by an automatic quadrature scheme. |
| url | http://psasir.upm.edu.my/id/eprint/24849/ http://psasir.upm.edu.my/id/eprint/24849/ http://psasir.upm.edu.my/id/eprint/24849/1/Approximating%20Cauchy.pdf |