Quadrature rule for Abel’s equations: Uniformly approximating fractional derivatives

Abstract

An automatic quadrature method is presented for approximating fractional derivative D q f (x) of a given function f (x), which is defined by an indefinite integral involving f (x). The present method interpolates f (x) in terms of the Chebyshev polynomials in the range [0, 1] to approximate the fractional derivative D q f (x) uniformly for 0 ≤ x ≤ 1, namely the error is bounded independently of x. Some numerical examples demonstrate the performance of the present automatic method.

Spectral approximations to the fractional integral and derivative

“…Li et al [10] also developed numerical algorithms based on the piecewise polynomial interpolation to approximate the fractional integral and the Caputo derivative, and to solve fractional differential equations. An automatic quadrature method based on the Chebyshev polynomials was presented for approximating the Caputo derivative in [26]. Some other computational schemes, such as the L1, L2 and L2C schemes, etc., are also introduced [8,11,13,14,16,18,19,21,23,25,28,29,31].…”

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].…”

“…As one can see, each curve is approximately a translation of sin(t) by a distance qπ/2 to the left. The exact value of the fractional derivative of this function is given by [39,41] …”

“…We compute 0 D q t f (t) for f (t) = (t+ν) q−1 and t ∈ [0, 1]. The exact value of the fractional derivative of this function is given by [39] The actual maximum error E q n , defined by (6.3), is plotted in Figure 5 in a semi-log coordinate system as a function of n for q = 1/2 and four values of the parameter ν (= 0.1, 0.2, 0.5 and 1.0). As we can see that E q n decreases rapidly.…”

Uniform approximation to fractional derivatives of functions of algebraic singularity

“…Let f (s) be a given function for s ∈ [0, 1]. The fractional derivatives in the Riemann-Liouville version D q f (s) and the Caputo version D q * f (s) (0 < q < 1) are defined by, respectively, with a relation between them [16] D q f (s)…”