An approximation method for high-order fractional derivatives of algebraically singular functions

Abstract

a r t i c l e i n f o Keywords: Fractional derivative of high order Algebraic singularity Quadrature rule Chebyshev interpolation Error analysis Uniform approximation a b s t r a c tThe fractional derivative D q f (s) (0 ≤ s ≤ 1) of a given function f (s) with a positive noninteger q is defined in terms of an indefinite integral. We propose a uniform approximation scheme to D q f (s) for algebraically singular functions f (s) = s α g(s) (α > −1) with smooth functions g(s). The present method consists of interpolating g(s) at sample points t j in [0, 1] by a finite sum of the Chebyshev polynomials. We demonstrate that for the non-negative integer m such that m < q < m + 1, the use of high-order derivatives g (i) (0) and g (i) (1) (0 ≤ i ≤ m) at both ends of [0, 1] as well as g(t j ), t j ∈ [0, 1] in interpolating g(s), is essential to uniformly approximate D q {s α g(s)} for 0 ≤ s ≤ 1 when α ≥ q − m − 1. Some numerical examples in the simplest case 1 < q < 2 are included.