An approximation method for high-order fractional derivatives of algebraically singular functions |
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Authors: | Takemitsu Hasegawa |
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Affiliation: | a Department of Information Science, University of Fukui, Fukui, 910-8507, Japanb Department of Systems Design and Engineering, Nanzan University, Seto, Aichi, 489-0863, Japan |
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Abstract: | The fractional derivative Dqf(s) (0≤s≤1) of a given function f(s) with a positive non-integer q is defined in terms of an indefinite integral. We propose a uniform approximation scheme to Dqf(s) for algebraically singular functions f(s)=sαg(s) (α>−1) with smooth functions g(s). The present method consists of interpolating g(s) at sample points tj in 0,1] by a finite sum of the Chebyshev polynomials. We demonstrate that for the non-negative integer m such that m<q<m+1, the use of high-order derivatives g(i)(0) and g(i)(1) (0≤i≤m) at both ends of 0,1] as well as g(tj), tj∈0,1] in interpolating g(s), is essential to uniformly approximate Dq{sαg(s)} for 0≤s≤1 when α≥q−m−1. Some numerical examples in the simplest case 1<q<2 are included. |
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Keywords: | Fractional derivative of high order Algebraic singularity Quadrature rule Chebyshev interpolation Error analysis Uniform approximation |
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