Approximating Cauchy-type singular integral by an automatic quadrature scheme

An automatic quadrature scheme is developed for the approximate evaluation of the product-type indefinite integral Q ( f , x , y , c ) = ∫ x y ρ ( t ) K ( t , c ) f ( t ) d t , − 1 ≤ x , y ≤ 1 , − 1 < c < 1 , where ρ ( t ) = 1 / 1 − t 2 , K ( t , c ) = 1 / ( t − c ) and f ( t ) is assumed to be a smooth function. In constructing an automatic quadrature scheme, we consider two cases: (1) − 1 < x < y < 1 , and (2) x = − 1 , y = 1 . In both cases the density function f ( t ) is replaced by the truncated Chebyshev polynomial p N ( t ) of the first kind of degree N . The approximation p N ( t ) yields an integration rule Q N ( f , x , y , c ) to the integral Q ( f , x , y , c ) . Interpolation conditions are imposed to determine the unknown coefficients of the Chebyshev polynomials p N ( t ) . Convergence problem of the approximate method is discussed in the classes of function C N + 1 , α [ − 1 , 1 ] and L p w [ − 1 , 1 ] . Numerically, it is found that when the singular point c either lies in or outside the interval ( x , y ) or comes closer to the end points of the interval [ − 1 , 1 ] , the proposed scheme gives a very good agreement with the exact solution. These results in the line of theoretical findings.