Cover Image

Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials

In this thesis, an automatic quadrature scheme is presented for evaluating the product type indefinite integral  Q(f,x,y,c)=y/x w (t) K (c;t) f (t) dt,-1 < x , y < 1,-1 < c < 1 where w ( t ) =1/ 1-t2 , k (c , t ) =1 /( t - c ) and f ( t ) is assumed to be a smooth function. In con...

Full description

Saved in:
Bibliographic Details
Main Author: Jamaludin, Nur Amalina
Format: Thesis
Language:English
English
Published: 2010
Online Access:http://psasir.upm.edu.my/id/eprint/12426/1/FS_2010_8A.pdf
http://psasir.upm.edu.my/id/eprint/12426/
Tags: Add Tag
No Tags, Be the first to tag this record!
id my.upm.eprints.12426
record_format eprints
spelling my.upm.eprints.124262013-05-27T07:52:10Z http://psasir.upm.edu.my/id/eprint/12426/ Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials Jamaludin, Nur Amalina In this thesis, an automatic quadrature scheme is presented for evaluating the product type indefinite integral  Q(f,x,y,c)=y/x w (t) K (c;t) f (t) dt,-1 < x , y < 1,-1 < c < 1 where w ( t ) =1/ 1-t2 , k (c , t ) =1 /( t - c ) and f ( t ) is assumed to be a smooth function. In constructing an automatic quadrature scheme for the case -1 < x < y < 1 the density function f ( t ) is approximated by the truncated Chebyshev polynomial   PN ( t ) of the first kind of degree N. The approximation  PN ( t ) yields an integration rule  Q( PN ,x , y , c ) to the integral Q ( f , x, y,c ). An automatic quadrature scheme for the case x- - 1,y - 1can easily be constructed by replacing f ( t )  with PN ( t ) and using the known formula /1-1 Tk ( t ) 1- t2 ( t - c ) dt =Uk - 2 ( c ) ,k = 1 ,.., N . In both cases the interpolation conditions are imposed to determine the unknown coefficients of the Chebyshev polynomials PN ( t ). The evaluations of Q( f , x , y,c ) ~= Q(PN x , y , c )for the set ( x,y,c ) can be efficiently computed by using backward direction method. The estimation of errors for an automatic quadrature scheme are obtained and convergence problem are discussed in the classes of functions CN + 1, a [-11] and LWP [-11]. The C code is developed to obtain the numerical results and they are presented and compared with the exact solution of SI for different functions f ( t ) . Numerical experiments are presented to show the efficiency and the accuracy of the method. It asserts the theoretical results. 2010-04 Thesis NonPeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/12426/1/FS_2010_8A.pdf Jamaludin, Nur Amalina (2010) Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials. Masters thesis, Universiti Putra Malaysia. English
institution Universiti Putra Malaysia
building UPM Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Putra Malaysia
content_source UPM Institutional Repository
url_provider http://psasir.upm.edu.my/
language English
English
description In this thesis, an automatic quadrature scheme is presented for evaluating the product type indefinite integral  Q(f,x,y,c)=y/x w (t) K (c;t) f (t) dt,-1 < x , y < 1,-1 < c < 1 where w ( t ) =1/ 1-t2 , k (c , t ) =1 /( t - c ) and f ( t ) is assumed to be a smooth function. In constructing an automatic quadrature scheme for the case -1 < x < y < 1 the density function f ( t ) is approximated by the truncated Chebyshev polynomial   PN ( t ) of the first kind of degree N. The approximation  PN ( t ) yields an integration rule  Q( PN ,x , y , c ) to the integral Q ( f , x, y,c ). An automatic quadrature scheme for the case x- - 1,y - 1can easily be constructed by replacing f ( t )  with PN ( t ) and using the known formula /1-1 Tk ( t ) 1- t2 ( t - c ) dt =Uk - 2 ( c ) ,k = 1 ,.., N . In both cases the interpolation conditions are imposed to determine the unknown coefficients of the Chebyshev polynomials PN ( t ). The evaluations of Q( f , x , y,c ) ~= Q(PN x , y , c )for the set ( x,y,c ) can be efficiently computed by using backward direction method. The estimation of errors for an automatic quadrature scheme are obtained and convergence problem are discussed in the classes of functions CN + 1, a [-11] and LWP [-11]. The C code is developed to obtain the numerical results and they are presented and compared with the exact solution of SI for different functions f ( t ) . Numerical experiments are presented to show the efficiency and the accuracy of the method. It asserts the theoretical results.
format Thesis
author Jamaludin, Nur Amalina
spellingShingle Jamaludin, Nur Amalina
Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials
author_facet Jamaludin, Nur Amalina
author_sort Jamaludin, Nur Amalina
title Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials
title_short Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials
title_full Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials
title_fullStr Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials
title_full_unstemmed Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials
title_sort automatic quadrature scheme for evaluating singular integral with cauchy kernel using chebyshev polynomials
publishDate 2010
url http://psasir.upm.edu.my/id/eprint/12426/1/FS_2010_8A.pdf
http://psasir.upm.edu.my/id/eprint/12426/
_version_ 1643825035808342016
score 13.211869

Similar Items