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

An automatic quadrature method is presented for approximating fractional derivative D q f ( 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 q f ( 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.