Uniform approximation to fractional derivatives of functions of algebraic singularity

Abstract

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

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

“…This is a generalization of our previous method [7] for 0 < q < 1, which could treat only the case m = 0 in (4). A simple extension of the method [7] would fail to approximate D q {s α g(s)} (m ≥ 1) uniformly.…”

“…Few references, however, appear to exist on uniform approximation schemes for a certain interval of s except for our recent papers [7,8] for 0 < q < 1. Particularly, paper [8] is for smooth functions f (s) while in [7] we treat the algebraically singular function f (s) = s α g(s) (α > −1), which is one of a practically important class of functions, with g(s)…”

“…A simple extension of the method [7] would fail to approximate D q {s α g(s)} (m ≥ 1) uniformly. Although in [7] we use Lagrange interpolation of g(t) on the Chebyshev nodes t j = (1 + cos π j/n)/2 (0 ≤ j ≤ n), this interpolation is not enough for the uniform approximation of D q {s α g(s)}, m ≥ 1.…”

Nonstandard Gauss—Lobatto quadrature approximation to fractional derivatives

“…[8,9,11,3,10,35]), there seems to exist a few literature on automatic quadrature for the fractional derivatives, see e.g. [20,24,39].…”

“…Problems with algebraic or/and logarithmic singularities can be solved using Müntz systems and quadratures of this type (cf. [30,32,11]) or using a procedure proposed in [20].…”