What is a fractional derivative?

By Manuel Duarte Ortigueira, J. Machado. Journal of Computational Physics, 2015

491 citations recorded by the source corpus.

This paper discusses the concepts underlying the formulation of operators capable of being interpreted as fractional derivatives or fractional integrals. Two criteria for required by a fractional operator are formulated. The Grunwald-Letnikov, Riemann-Liouville and Caputo fractional derivatives and the Riesz potential are accessed in the light of the proposed criteria. A Leibniz rule is also obtained for the Riesz potential.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/e33f6a013d5aa294a2c90bc01bc46d40bda914ea

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In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

Newly Defined Conformable Derivatives (2015, background): “However, it lacks some of the agreed upon properties for fractional derivatives [18].”
Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for generalized fractional integrals ☆ (2016, methodology): “tives, respectively. For consistency, we use the same name for those operators in question. It can be shown that the derivatives in question satisfy the fractional derivative criteria (test) given in [20, 32]. These operators have applications in ﬁelds such as in probability theory [1], theory of in”
Fractional derivatives and the fundamental theorem of fractional calculus (2020, unknown): “he ”new fractional integrals and derivatives” ( [6,8,20,24–26]), the integro-diﬀerential operators of convolution type with some general kernels ( [8,14,15,24,27,28]), and the abstract axioms of FC ( [11,19,22]). In this framework, the ”classical” deﬁnitions of the fractional integrals and derivativ”
On a Caputo-type fractional derivative (2018, methodology): “A natural continuation of this paper is to show the validity of a Leibniz-type rule in order to confirm, as proposed by Ortigueira and Tenreiro Machado [14], that our operator is really a fractional operator.”
Desiderata for Fractional Derivatives and Integrals (2019, background): “Little or no attention is given to a domain of definition for the fractional derivatives in [2].”
Three-Point Boundary Value Problems for Conformable Fractional Differential Equations (2015, background): “Some authors (see [7, 8]) have argued that conformable fractional derivative is not a truly fractional operator.”
Stability and existence results for a class of nonlinear fractional differential equations with singularity (2018, unknown): “With the help of (11) and (12), we get β(t, s) ≥ 0, for s, t ∈ (0, 1).”