Uniform approximation to fractional derivatives of functions of algebraic singularity

✍ Scribed by Takemitsu Hasegawa; Hiroshi Sugiura

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 528 KB
• Volume: 228
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2008.09.018

✦ Synopsis

Five-term recurrence relation a b s t r a c t 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.