Quadrature rule for indefinite integral of algebraic-logarithmic singular integrands

概要

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An automatic quadrature method is presented for approximating the indefinite integral of functions having algebraic–logarithmic singularities Q(x,y,c;f)=∮yx f(t)|t-c|^αlog|t-c|dt,-1<_x,y,c<_1,α>-1,within a finite range [-1,1] for some smooth function f(t), that is approximated by a finite sum of Chebyshev polynomials. We expand the given indefinite integral in terms of Chebyshev polynomials by using auxiliary algebraic–logarithmic functions. Present scheme approximates the indefinite integral Q(x,y,c;f) uniformly, namely bounds the approximation error independently of the value c as well x and y. This fact enables us to evaluate the integral transform Q(x,y,c;f) with varied values of x, y and c efficiently. Some numerical examples illustrate the performance of the present quadrature scheme.