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

The fractional derivative Dqf(s)Dqf(s) (0≤s≤10≤s≤1) of a given function f(s)f(s) with a positive non-integer qq is defined in terms of an indefinite integral. We propose a uniform approximation scheme to Dqf(s)Dqf(s) for algebraically singular functions f(s)=sαg(s)f(s)=sαg(s) (α>−1α>−1) with smooth functions g(s)g(s). The present method consists of interpolating g(s)g(s) at sample points tjtj in [0,1][0,1] by a finite sum of the Chebyshev polynomials. We demonstrate that for the non-negative integer mm such that m