Approximating Cauchy-type singular integral by an automatic quadrature scheme

Abstract

An automatic quadrature scheme is developed for the approximate evaluation of the product-type indefinite integral Q(f,x,y,c)=∫xyρ(t)K(t,c)f(t)dt,−1≤x,y≤1,−1<c<1, where ρ(t)=1/1−t2, 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 pN(t) of the first kind of degree N. The approximation pN(t) yields an integration rule QN(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 pN(t). Convergence problem of the approximate method is discussed in the classes of function CN+1,α[−1,1] and Lpw[−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.