Uniform approximation to fractional derivatives of functions of algebraic singularity

Fractional derivative Dqf(x) (0−1) with g(x) being a well-behaved function, we propose a quadrature method for uniformly approximating Dq{xαg(x)}. The 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∈[0,1]. Numerical examples demonstrate the performance of the present method.