A comment on some new definitions of fractional derivative

Abstract

After reviewing the definition of two differential operators which have been recently introduced by Caputo and Fabrizio and, separately, by Atangana and Baleanu, we present an argument for which these two integro-differential operators can be understood as simple realizations of a much broader class of fractional operators, i.e. the theory of Prabhakar fractional integrals. Furthermore, we also provide a series expansion of the Prabhakar integral in terms of Riemann-Liouville integrals of variable order. Then, by using this last result we finally argue that the operator introduced by Caputo and Fabrizio cannot be regarded as fractional. Besides, we also observe that the one suggested by Atangana and Baleanu is indeed fractional, but it is ultimately related to the ordinary Riemann-Liouville and Caputo fractional operators. All these statements are then further supported by a precise analysis of differential equations involving the aforementioned operators. To further strengthen our narrative, we also show that these new operators do not add any new insight to the linear theory of viscoelasticity when employed in the constitutive equation of the Scott-Blair model.

10 pages, 1 figure, to appear in Nonlinear Dynamics, comment added

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• No Nonlocality. No Fractional Derivative
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Cited by in corpus (19)

• A practical guide to Prabhakar fractional calculus
• On fractional calculus with general analytic kernels
• Why fractional derivatives with nonsingular kernels should not be used
• Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions
• A comment on a controversial issue: a Generalized Fractional Derivative cannot have a regular kernel
• A critical analysis of the conformable derivative
• Fractional derivatives and the fundamental theorem of Fractional Calculus
• General fractional calculus and Prabhakar's theory
• Modeling biological systems with an improved fractional Gompertz law
• Scott-Blair models with time-varying viscosity
• Stability of fractional-order systems with Prabhakar derivatives
• Storage and dissipation of energy in Prabhakar viscoelasticity
• Neglecting nonlocality leads to unreliable numerical methods for fractional differential equations
• Dispersion relations for the time-fractional Cattaneo-Maxwell heat equation
• Biased continuous-time random walks with Mittag-Leffler jumps
• Time-fractional Caputo derivative versus other integro-differential operators in generalized Fokker-Planck and generalized Langevin equations
• On complete monotonicity of solution to the fractional relaxation equation with the $n$th level fractional derivative
• Remarks about the existence of conformable derivatives and some consequences
• On the mistake in defining fractional derivative using a non-singular kernel