Quadrature rule for Abel's equations: Uniformly approximating fractional derivatives

Authors:
Hiroshi Sugiura;Takemitsu Hasegawa
Affiliations:
Department of Information Systems and Mathematical Sciences, Nanzan University, Seto, Aichi, 489-0863, Japan;Department of Information Science, University of Fukui, Fukui, 910-8507, Japan
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

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Abstract

An automatic quadrature method is presented for approximating fractional derivative D^qf(x) of a given function f(x), which is defined by an indefinite integral involving f(x). The present method interpolates f(x) in terms of the Chebyshev polynomials in the range [0, 1] to approximate the fractional derivative D^qf(x) uniformly for 0@?x@?1, namely the error is bounded independently of x. Some numerical examples demonstrate the performance of the present automatic method.