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

The fractional derivative D(q)f (s) (0 <= s <= 1) of a given function f (s) with a positive non-integer 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(alpha)g(s) (alpha > -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) is an element of [0, 1] in interpolating g (s), is essential to uniformly approximate D(q) {s(alpha)g(s)} for 0 <= s <= 1 when alpha >= q - m - 1. Some numerical examples in the simplest case 1 < q < 2 are included. (C) 2011 Elsevier Ltd. All rights reserved.