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The flaw in the conformable calculus: it is conformable because it is not fractional

Ahmed A. Abdelhakim

Abstract

We point out a major flaw in the conformable calculus. We demonstrate why it fails at defining a fractional derivative and where exactly these tempting conformability properties come from.

The flaw in the conformable calculus: it is conformable because it is not fractional

Abstract

We point out a major flaw in the conformable calculus. We demonstrate why it fails at defining a fractional derivative and where exactly these tempting conformability properties come from.

Paper Structure

This paper contains 3 sections, 8 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The problems in the statements and proofs in khalil
  3. The problems in the statements and proofs in Abdeljawad

Key Result

Theorem 1

(ahmedMachado, Theorem 1) Fix $\,0<\alpha<1$ and let $t>0$. A function $\,f:[0,\infty[\,\longrightarrow \mathbb{R}\,$ has a "conformable fractional derivative" of order $\alpha$ at $t$ if and only if it is differentiable at $t$, in which case we have the pointwise relation

Figures (2)

  • Figure 1: The behavior of $T^{0}_{\alpha}g_{1}$ is very different from that of $D^{\alpha}_{0^{+}}g_{1}$, ${^C}D^{\alpha}_{0^{+}}g_{1}$
  • Figure 2: The behavior of $T^{0}_{\alpha}g_{2}$ is very different from that of $D^{\alpha}_{0^{+}}g_{2}$, ${^C}D^{\alpha}_{0^{+}}g_{2}$

Theorems & Definitions (25)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • ...and 15 more