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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| 主要作者: | |
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| 格式: | Thesis |
| 语言: | English English |
| 出版: |
2010
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| 主题: | |
| 在线阅读: | http://psasir.upm.edu.my/id/eprint/12426/1/FS_2010_8A.pdf |
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| 总结: | 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. |
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