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 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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