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...

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Main Author:	Jamaludin, Nur Amalina
Format:	Thesis
Language:	English English
Published:	2010
Subjects:	Numerical integration Cauchy problem - Numerical solutions Cauchy integrals
Online Access:	http://psasir.upm.edu.my/id/eprint/12426/1/FS_2010_8A.pdf
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spelling	my-upm-ir.124262013-05-27T07:52:10Z Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials 2010-04 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. Numerical integration Cauchy problem - Numerical solutions Cauchy integrals 2010-04 Thesis http://psasir.upm.edu.my/id/eprint/12426/ http://psasir.upm.edu.my/id/eprint/12426/1/FS_2010_8A.pdf application/pdf en public masters Universiti Putra Malaysia Numerical integration Cauchy problem - Numerical solutions Cauchy integrals Faculty Of Science English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Numerical integration Cauchy problem - Numerical solutions Cauchy integrals
spellingShingle	Numerical integration Cauchy problem - Numerical solutions Cauchy integrals Jamaludin, Nur Amalina Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials
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
qualification_level	Master's degree
author	Jamaludin, Nur Amalina
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
granting_institution	Universiti Putra Malaysia
granting_department	Faculty Of Science
publishDate	2010
url	http://psasir.upm.edu.my/id/eprint/12426/1/FS_2010_8A.pdf
_version_	1747811363495870464

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

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