Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 279681, 8 pages http://dx.doi.org/10.1155/2013/279681 Research Article A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions Abdon Atangana 1 and Aydin Secer 2 1 Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa 2 Yildiz Technical University, Department of Mathematical Engineering, Davutpasa, 34210 ˙ Istanbul, Turkey Correspondence should be addressed to Aydin Secer; asecer@yildiz.edu.tr Received 10 January 2013; Accepted 1 March 2013 Academic Editor: Mustafa Bayram Copyright © 2013 A. Atangana and A. Secer. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te purpose of this note is to present the diferent fractional order derivatives defnition that are commonly used in the literature on one hand and to present a table of fractional order derivatives of some functions in Riemann-Liouville sense On other the hand. We present some advantages and disadvantages of these fractional derivatives. And fnally we propose alternative fractional derivative defnition. 1. Introduction Fractional calculus has been used to model physical and engineering processes, which are found to be best described by fractional diferential equations. It is worth nothing that the standard mathematical models of integer-order deriva- tives, including nonlinear models, do not work adequately in many cases. In the recent years, fractional calculus has played a very important role in various felds such as mechanics, electricity, chemistry, biology, economics, notably control theory, and signal and image processing. Major topics include anomalous difusion, vibration and control, continu- ous time random walk, Levy statistics, fractional Brownian motion, fractional neutron point kinetic model, power law, Riesz potential, fractional derivative and fractals, computa- tional fractional derivative equations, nonlocal phenomena, history-dependent process, porous media, fractional flters, biomedical engineering, fractional phase-locked loops, frac- tional variational principles, fractional transforms, fractional wavelet, fractional predator-prey system, sof matter mechan- ics, fractional signal and image processing; singularities analysis and integral representations for fractional diferential systems; special functions related to fractional calculus, non- Fourier heat conduction, acoustic dissipation, geophysics, relaxation, creep, viscoelasticity, rheology, fuid dynamics, chaos and groundwater problems. An excellent literature of this can be found in [1–9]. Tese entire models are making use of the fractional order derivatives that exist in the literature. However, there are many of these defnitions in the literature nowadays, but few of them are commonly used, including Riemann-Liouville [10, 11], Caputo [5, 12], Weyl [10, 11, 13], Jumarie [14, 15], Hadamard [10, 11], Davison and Essex [16], Riesz [10, 11], Erdelyi-Kober [10, 11], and Coimbra [17]. All these fractional derivatives defnitions have their advantages and disadvantages. Te purpose of this note is to present the result of fractional order derivative for some function and from the results establish the disadvantages and advantages of these fractional order derivative defnitions. We shall start with the defnitions. 2. Definitions Tere exists a vast literature on diferent defnitions of frac- tional derivatives. Te most popular ones are the Riemann- Liouville and the Caputo derivatives. For Caputo we have  0    ( ())= 1 Γ(−) ∫  0 ( − ) −−1   ()   , −1<≤. (1)