Approximating Cauchy-type singular integral by an automatic quadrature scheme
- Eskhuvatov, Z.K.
- Ahmedov, A.
- Nik Long, N.M.A.
- Amalina, N.J.
DOIAbstract
AbstractAn 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 cl...
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