Why fractional derivatives with nonsingular kernels should not be used
By Kai Diethelm, Roberto Garrappa, Andrea Giusti, Martin Stynes. Fractional Calculus and Applied Analysis, 2020
142 citations recorded by the source corpus.
Abstract In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.
Subfields: Control and Systems Engineering, Modeling and Simulation, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/46b57d6c888774c139e94a887b7845481004be41
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Citations in the corpus
- Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense (2022, background): “However, merely condition (15) does not ensure an essential requirement for the kernel k of the GFD, namely, the condition that this kernel should have a singularity at the origin, see [43, 44, 46].”
- General Fractional Integrals and Derivatives with the Sonine Kernels (2021, background): “On the other hand, the kernels of the fractional integrals and derivatives should be singular ( [27]).”
- Fractional derivatives and the fundamental theorem of fractional calculus (2020, background): “These discussions mainly concerned the ”new fractional integrals and derivatives” ( [6,8,11,25,29–31]), the integro-differential operators of convolution type with some general kernels ( [6, 11, 17, 18, 29, 32, 33]), and the abstract axioms of FC ( [14, 24, 27]).”
- Operational Calculus for the General Fractional Derivative and Its Applications (2021, unknown): “On the other hand, the kernels of any fractional integral and derivative should be singular (see [5] for a detailed discussion of this topic).”