Quadrature rule for indefinite integral of algebraic-logarithmic singular integrands

Authors:
Takemitsu Hasegawa;Hiroshi Sugiura
Affiliations:
Department of Information Science, University of Fukui, Fukui 910-8507, Japan;Department of Mathematical Sciences, Nanzan University, Seto, Aichi 489-0863, Japan
Venue:
Journal of Computational and Applied Mathematics
Year:
2007

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Abstract

An automatic quadrature method is presented for approximating the indefinite integral of functions having algebraic-logarithmic singularities Q(x,y,c;f)=@!"x^yf(t)|t-c|^@alog|t-c|dt, -1=-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.