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

An automatic quadrature method is presented for approximating fractional derivative Dqf(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 Dqf(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.