Uniform approximation to fractional derivatives of functions of algebraic singularity

Fractional derivative D q f ( 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 α g ( x ) ( α > − 1 ) with g ( x ) being a well-behaved function, we propose a quadrature method for uniformly approximating D q { x α g ( 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.