MSC: 65D32 45E05 65R20 Keywords:An automatic quadrature scheme Product integral Singular integral Clenshaw-Curtis rules Chebyshev polynomials Indefinite integral Recurrence relation a b s t r a c t An automatic quadrature scheme is developed for the approximate evaluation of the product-type indefinite integraland 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 w p [−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.Crown

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Hasegawa constructed the automatic quadrature scheme (AQS), of Cauchy principle value integrals for smooth functions. There is a close connection between Hadamard and Cauchy principle value integral. In this paper, we modify AQS for hypersingular integrals with second-order singularities, using hasegawa's formula and based on the relations between Hadamard finite part integral and Cauchy principle value integral. Numerical experiments are also given, to validate the modified AQS.