A New Approach to Generalized Fractional Derivatives

The author \mbox{(Appl. Math. Comput. 218(3):860-865, 2011)} introduced a new fractional integral operator given by, $\big({}^ρ\mathcal{I}^α_{a+}f\big)(x) = \frac{ρ^{1- α}}{Γ(α)} \int^x_a \frac{τ^{ρ-1} f(τ) }{(x^ρ- τ^ρ)^{1-α}}\, dτ,$ 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.

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