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Approximations of fractional integrals and Caputo fractional derivatives. (English) Zbl 1101.65028
In a series of recent papers [see {\it K. Diethelm, A. D. Freed} and {\it N. J. Ford}, Numer. Algorithms 36, No. 1, 31--52 (2004; Zbl 1055.65098)], and the references cited therein], the reviewer and his collaborators have proposed and analysed a numerical scheme for the approximation of $J^\alpha$, the Riemann-Liouville fractional integral of order $\alpha$. The method is based on a product trapezoidal quadrature formula. In the paper under review, this approximation operator $J_h^\alpha$ is recalled. Then the author turns the attention towards the numerical computation of the Caputo fractional derivative $D^\alpha_* := J^{m-\alpha} D^m$, where $m = \lceil \alpha \rceil$ and $D^m$ is the classical differential operator of order $m$. For this problem, he suggests a straightforward modification of the idea described above, namely to use $J_h^{m-\alpha} D^m$ as an approximation for $D^\alpha_*$. Obviously, the error analysis can be carried over directly. It is evident that the proposed algorithm requires the evaluation of derivatives of the underlying function. The examples presented in the paper are restricted to cases where such information is very easily obtained.

65D25Numerical differentiation
26A33Fractional derivatives and integrals (real functions)
65D32Quadrature and cubature formulas (numerical methods)
Full Text: DOI
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