Letter to the editor
A critical analysis of the Caputo–Fabrizio operator

https://doi.org/10.1016/j.cnsns.2017.12.001Get rights and content

Highlights

  • The note analyses the Caputo-Fabrizio operator using the Laplace transform and well-known system tools.

  • Some simulation results, namely the Bode diagrams computation, show clearly that the operator is not fractional. The time simulation help in showing it is not a derivative.

Abstract

This short note analyses the Caputo–Fabrizio operator. It is verified that it does not fit the usual concepts neither for fractional nor for integer derivative/integral.

Introduction

In recent years, several operators were proposed having the keyword “fractional”. One of the proposals consists of the Caputo-Fabrizio (CF) operator [1] that is defined by the expression Dt(α)f(t)=M(α)1αatf(τ)eα1α(tτ)dτ,where 0 ≤ α ≤ 1, a(,t) and M(α) is a normalization function such that M(0)=M(1)=1. We will assume that (1) is valid for functions having Laplace transform (LT).

In the following we analyse this operator and we verify that it implements an integer order highpass filter. We can prove that the CF operator is neither fractional, nor a derivative, by showing that it does not verify the Leibniz rule [11]. The proof is somehow involved and we shall adopt here a different perspective based on Laplace transform and Bode diagrams.

Section snippets

Is the CF operator fractional?

Let us consider expression (1) and a=0 just in order to simplify expressions. Recalling that the multiplication of 2 Laplace transforms corresponds to the convolution in the time domain, Eq. (1) can be written as g(t)=Dt(α)f(t)=M(α)1αf(t)*[eα1αtɛ(t)],where ε(t) stands for the Heaviside unit step and * denotes the convolution.

The bilateral Laplace transform (no initial conditions are needed [9]) of (2) gives G(s)=M(α)1αss+α1αF(s),where F(s)=L[f(t)] and G(s)=L[g(t)]. Eq. (3) can be

Conclusions

The analysis of the Caputo–Fabrizio operator revealed that it is neither a fractional nor a derivative operator [6]. Similarly, the generalization of the operator is not a fractional derivative.

In this short note was adopted a strategy based on Laplace transform and related properties. The results support the use in FC of mathematical tools well established in the area of applied sciences.

Conflict of Interest

The authors declare that they have no conflict of interest.

Acknowledgements

This work was partially funded by National Funds through the Foundation for Science and Technology of Portugal, under the projects PEst-UID/EEA/00066/2013.

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