A New Approach to Generalized Fractional Derivatives
Abstract
The author \mbox{(Appl. Math. Comput. 218(3):860-865, 2011)} introduced a new fractional integral operator given by, \[ \big({}^\rho \mathcal{I}^\alpha_{a+}f\big)(x) = \frac{\rho^{1- \alpha }}{\Gamma({\alpha})} \int^x_a \frac{\tau^{\rho-1} f(\tau) }{(x^\rho - \tau^\rho)^{1-\alpha}}\, d\tau, \] 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2011
- DOI:
- 10.48550/arXiv.1106.0965
- arXiv:
- arXiv:1106.0965
- Bibcode:
- 2011arXiv1106.0965K
- Keywords:
-
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Combinatorics;
- 26A33
- E-Print:
- 12 Pages, 2 Figures