Uniform approximation to fractional derivatives of functions of algebraic singularity

Fractional derivative D^qf(x) (0 < q < 1, 0 <_ _ - x <_ _ - 1) of a function f(x) is defined in terms of an indefinite integral involving f(x). For functions of algebraic singularity f(x) = x^αg(x) (α > -1) with g(x) being a well-behaved function, we propose a quadrature method for uniformly approximating D^q{x^αg(x)g}. Present method consists of interpolating g(x) at abscissae in [0,1] by a finite sum of Chebyshev polynomials. It is shown that the use of the lower endpoint x = 0 as an abscissa is essential for the uniform approximation, namely to bound the approximation errors independently of x 2 [0,1]. Numerical examples demonstrate the performance of the present method