Citation
Jamaludin, Nur Amalina
(2010)
Automatic Quadrature Scheme For Evaluating Singular Integral With Cauchy Kernel Using Chebyshev Polynomials.
Masters thesis, Universiti Putra Malaysia.
Abstract
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/ 1t2 , 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
/11 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.
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Additional Metadata
Item Type: 
Thesis
(Masters)

Subject: 
Numerical integration 
Subject: 
Cauchy problem  Numerical solutions 
Subject: 
Cauchy integrals 
Call Number: 
FS 2010 8 
Chairman Supervisor: 
Zainidin K. Eshkuvatov, PhD 
Divisions: 
Faculty of Science 
Depositing User: 
Mohd Nezeri Mohamad

Date Deposited: 
12 Jul 2011 01:40 
Last Modified: 
27 May 2013 07:52 
URI: 
http://psasir.upm.edu.my/id/eprint/12426 
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