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A new approach to generalized fractional derivatives. (English) Zbl 1317.26008
Summary: In [Appl. Math. Comput. 218, No. 3, 860–865 (2011; Zbl 1231.26008)], we introduced a new fractional integral operator given by \[ (\rho \mathcal I^{\alpha}_{a+}f)(x) = \frac {\rho^{1-\alpha}}{\Gamma(\alpha)} \int^x_a \frac {\tau^{\rho-1}f(\tau)}{(x^{\rho}-\tau^{\rho})^{1-\alpha}}d\tau, \] which generalizes the well-known Riemann-Liouville and the Hadamard fractional integrals. In this paper, we present a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivatives to a single form. We also obtain two representations of the generalized derivative in question. An example is given to illustrate the results.

26A33 Fractional derivatives and integrals
65R10 Numerical methods for integral transforms
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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