A new approach to generalized fractional derivatives. (English) Zbl 1317.26008

Katugampola, Udita N.

Bull. Math. Anal. Appl. 6, No. 4, 1-15 (2014).

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.

MSC:

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

Keywords:

fractional calculus; generalized fractional derivatives; Riemann-Liouville fractional derivative; Hadamard fractional derivative; Erdélyi-Kober operator; Taylor series expansion