On error estimation of automatic quadrature scheme for the evaluation of Hadamard integral of second order singularity.

This paper presents an automatic quadrature scheme (AQS) for the evaluation of hypersingular integrals (HSI) Qi(f, x) = ∫-1 1 wi(t)f(t)/(t - x)2 dt, x ∈ [-1, 1], i = 0, 1, 2, (1) where w0(t) = 1, w1(t) = √1 - t2, w2(t) = √1/1-t2 are the weights, and the given function f imperative to have certain smoothness or continuity properties. Particular attention is paid to error estimate of the developed AQS, where it shows the acquired AQS scheme is obtained in the class of functions CN+2,α[-1, 1] which converges to the exact very fast by increasing the knot points. The first and second kind of Chebyshev polynomials are used in the conjecture. Several numerical examples clearly demonstrate the developed AQS rendering efficient, accurate and reliable results. This research gives comparative performances of the present method with others.

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