A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions

By Abdon Atangana, Aydin Seçer. 2013

267 citations recorded by the source corpus.

The purpose of this note is to present the different fractional order derivatives definition 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 finally we propose alternative fractional derivative definition.

Subfields: Control and Systems Engineering, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/b1a7d89b3f971e2f2b3f7a53e307e1562c3dc8d2

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In Control and Systems Engineering

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

Extension of Matched Asymptotic Method to Fractional Boundary Layers Problems (2014, unknown): “x (f (x)) = lim ε→0 f(x + ε(x + 1/Γ (β)) 1−β ) − f (x) ε , (7) for all x ≥ a, β ∈ (0, 1].”
A Generalized Definition of the Fractional Derivative with Applications (2021, methodology): “Caputo derivatives are defined only for differentiable functions, while the functions that do not have first-order derivative may have fractional derivatives of all orders less than one in the Riemann–Liouville sense (see [13]).”
A NEW APPROACH TO GENERALIZED FRACTIONAL DERIVATIVES (2011, background): “1 cosnτ nα , and 0 ≤ τ ≤ 2π. Further properties of this operator can be found in [65,66] and the references therein. Jumarie proposed a simple modiﬁcation to the Riemann-Liouville derivative given by [3,26,27], Dα xf(x) = 1 Γ(n−α) dn dxn Zx 0 (x −τ)n−α−1 f(τ) −f(0) dτ, n −1 < α ≤ n. with the prop”
Numerical approximation of Riemann‐Liouville definition of fractional derivative: From Riemann‐Liouville to Atangana‐Baleanu (2018, background): “It worth noting that there exists only two main definition of fractional derivative, the first was proposed by Riemann and Liouville and is the derivative of the convolution of a given function and a power law kernel, the second one was suggested by Caputo and it is the convolution of the local deri”
The Use of Fractional Order Derivative to Predict the Groundwater Flow (2013, background): “The RiemannLiouville derivative of a constant is not zero while Caputo derivative of a constant is zero but demands higher conditions of regularity for differentiability [6–10]: to compute the fractional derivative of a function in the Caputo sense, we must first calculate its derivative [11].”