The Failure of Certain Fractional Calculus Operators in Two Physical Models

Abstract

The ability of the so-called Caputo-Fabrizio (CF) and Atangana-Baleanu (AB) operators to create suitable models for real data is tested with real world data. Two alternative models based on the CF and AB operators are assessed and compared with known models for data sets obtained from electrochemical capacitors and the human body electrical impedance. The results show that the CF and AB descriptions perform poorly when compared with the classical fractional derivatives.

A Comment on a Controversial Issue: A Generalized Fractional Derivative Cannot Have a Regular Kernel

“…Several paradoxes involving the GFD with regular kernels have been pointed out in [21,14,22,5]. In [15] it has been shown that the models involving the GFD with regular kernels poorly reflect the real world data. All these paradoxes stem from the fact the GFD with regular kernels are equivalent to Volterra integral operators, while the associated generalized fractional integrals are not integral operators.…”

The effect of fractional calculus on the formation of quantum‐mechanical operators

“…Abdelhakim has investigated a major flaw in the so‐called conformable calculus. The authors have shown that the Caputo–Fabrizio and Atangana–Baleanu descriptions perform poorly when compared with the classical fractional derivatives . In Sousa and de Oliveira, by means of the six‐parameter truncated Mittag–Leffler function, a new type of fractional derivative, which the authors called truncated V‐fractional derivative, is introduced.…”

On Controllability of Fractional Continuous-Time Systems

“…e authors in [28] showed that the ABC definition cannot be useful in modeling problems such as the fractional diffusion equation because the solutions obtained for these equations do not satisfy the initial condition. Ortigueira et al [29] showed that the models involving the generalized fractional derivative with regular kernels poorly reflect the real-world data. In responses to these criticisms, Atangana and Gómez-Aguilar [30] emphasized the need to account for a fractional calculus approach without an imposed index law and with nonsingular kernels.…”

“…Furthermore, Sabatier [31] showed that the papers [26,27] are not correct and produce the wrong conclusion on the restriction imposed by nonsingular kernels. In a comment written by Baleanu [32], it has been shown that the opinions of Ortigueira et al [29] are not consistent. Also, Atangana and Goufo [33] presented some interesting results to clarify the mistake and lack of understanding for those writing against derivatives with nonsingular kernels.…”