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A critical analysis of the conformable derivative

Ahmed A. Abdelhakim, José A. Tenreiro Machado

Abstract

We prove that conformable ``fractional" differentiability of a function $f:[0,\infty[\,\longrightarrow \mathbb{R}$ is nothing else than the classical differentiability. More precisely, the conformable $α$-derivative of $f$ at some point $x>0$, where $0<α<1$, is the pointwise product $x^{1-α}f^{\prime}(x)$. This proves the lack of significance of recent studies of the conformable derivatives. The results imply that interpreting fractional derivatives in the conformable sense alters fractional differential problems into differential problems with the usual integer-order derivatives that no longer describe the original fractional physical phenomena. A general fractional viscoelasticity model is analysed to illustrate this state of affairs. We also test the modelling efficiency of the conformable derivative using a fractional model of viscoelastic deformation of tight sandstone, and a fractional world population growth model.

A critical analysis of the conformable derivative

Abstract

We prove that conformable ``fractional" differentiability of a function is nothing else than the classical differentiability. More precisely, the conformable -derivative of at some point , where , is the pointwise product . This proves the lack of significance of recent studies of the conformable derivatives. The results imply that interpreting fractional derivatives in the conformable sense alters fractional differential problems into differential problems with the usual integer-order derivatives that no longer describe the original fractional physical phenomena. A general fractional viscoelasticity model is analysed to illustrate this state of affairs. We also test the modelling efficiency of the conformable derivative using a fractional model of viscoelastic deformation of tight sandstone, and a fractional world population growth model.

Paper Structure

This paper contains 10 sections, 4 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction and remarks
  2. A misleading counterexample and incorrect statements
  3. Main results
  4. Applications
  5. Comparative analysis of a fractional model
  6. Efficiency in modelling a real-world problem
  7. Gross National Production
  8. World population
  9. Conclusion
  10. Compliance with Ethical Standards

Key Result

theorem 1

Fix $\,0<\alpha<1$ and let $x>0$. A function $\,f:[0,\infty[\,\longrightarrow \mathbb{R}\,$ has a conformable fractional derivative of order $\alpha$ at $x$ if and only if it is differentiable at $x$ and (q1) holds.

Figures (4)

  • Figure 1: The profile of the strain that corresponds to an exponentially decaying stress on a viscoelastic material
  • Figure 2: The profile of the strain that corresponds to an oscillating stress on a viscoelastic material ($A_{1}=1$).
  • Figure 3: The strain that corresponds to an oscillating stress on a viscoelastic material ($b=A_{1}=1$).
  • Figure :

Theorems & Definitions (6)

  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • theorem 4