# How Many Fractional Derivatives Are There?

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## Abstract

**:**

## 1. Introduction

**Remark**

**1.**

- We work on $\mathbb{R}$.
- We use the two-sided Laplace transform (LT):$$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]=\underset{\mathbb{R}}{\overset{}{\int}}f\left(t\right){e}^{-st}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t,$$
- The Fourier transform (FT), $\mathcal{F}\left[f\left(t\right)\right]$, is obtained from the LT through the substitution $s=i\kappa ,$ with $\kappa \in \mathbb{R}.$

## 2. What Is a Fractional Derivative?

**Definition**

**1.**

**P**defined as:

**P1**- LinearityThe operator is linear.
- P2
- IdentityThe zero order derivative of a function returns the function itself.
- P3
- Backward compatibilityWhen the order is integer, FD gives the same result as the ordinary derivative.
- P4
- The index law$${D}^{\alpha}{D}^{\beta}f\left(t\right)={D}^{\alpha +\beta}f\left(t\right)$$
- P5
- The generalized Leibniz rule$${D}^{\alpha}\left[f\left(t\right)g\left(t\right)\right]=\sum _{i=0}^{\infty}\left(\genfrac{}{}{0pt}{}{\alpha}{i}\right){D}^{i}f\left(t\right){D}^{\alpha -i}g\left(t\right)$$

- P’4
- The generalized index law$${D}^{\alpha}{D}^{\beta}f\left(t\right)={D}^{\alpha +\beta}f\left(t\right)$$

- The anti-derivative is unique.
- The anti-derivative is a left and right inverse, while any primitive is only right inverse.

## 3. Unified Fractional Derivative

**Definition**

**2.**

- Fourier transformationIt was introduced above in (2). It permits obtaining (8) from (7), using the convolution theorem. It has another consequence:$${D}_{\theta}^{\alpha}f\left(t\right)=cos\left(\theta \frac{\pi}{2}\right){D}_{0}^{\alpha}f\left(t\right)+sin\left(\theta \frac{\pi}{2}\right){D}_{1}^{\alpha}f\left(t\right).$$
- EigenfunctionsLet $f\left(x\right)={e}^{i\kappa x},\phantom{\rule{0.277778em}{0ex}}\kappa ,x\in \mathbb{R}.$ Then,$${D}_{\theta}^{\beta}{e}^{i\kappa x}={\left|\kappa \right|}^{\beta}{e}^{i\frac{\pi}{2}\theta \xb7\mathrm{sgn}\left(\kappa \right)}{e}^{i\kappa x},$$
- Periodicity in $\theta $The UFD is periodic in $\theta $ with period 4$${D}_{\theta}^{\beta}f\left(x\right)={(-1)}^{n}{D}_{\theta +2n}^{\beta}f\left(x\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\in \mathbb{Z},$$
- Additivity and commutativity of the orders$${D}_{{\theta}_{1}}^{{\beta}_{1}}{D}_{{\theta}_{2}}^{{\beta}_{2}}f\left(x\right)={D}_{{\theta}_{1}+{\theta}_{2}}^{{\beta}_{1}+{\beta}_{2}}f\left(x\right).$$In particular, conjugating (13) with (12),$$\begin{array}{cc}\hfill {D}_{0}^{{\beta}_{1}}{D}_{0}^{{\beta}_{2}}f\left(x\right)& ={D}_{0}^{{\beta}_{1}+{\beta}_{2}}f\left(x\right),\hfill \end{array}$$$$\begin{array}{cc}\hfill {D}_{1}^{{\beta}_{1}}{D}_{1}^{{\beta}_{2}}f\left(x\right)& =-{D}_{0}^{{\beta}_{1}+{\beta}_{2}}f\left(x\right),\hfill \end{array}$$$$\begin{array}{cc}\hfill {D}_{0}^{{\beta}_{1}}{D}_{1}^{{\beta}_{2}}f\left(x\right)& ={D}_{1}^{{\beta}_{1}+{\beta}_{2}}f\left(x\right),\hfill \end{array}$$$$\begin{array}{cc}\hfill {D}_{{\beta}_{1}}^{{\beta}_{1}}{D}_{{\beta}_{2}}^{{\beta}_{2}}f\left(x\right)& ={D}_{{\beta}_{1}+{\beta}_{2}}^{{\beta}_{1}+{\beta}_{2}}f\left(x\right),\hfill \end{array}$$$$\begin{array}{cc}\hfill {D}_{\theta}^{\frac{\beta}{2}}{D}_{-\theta}^{\frac{\beta}{2}}f\left(x\right)& ={D}_{0}^{\beta}f\left(x\right),\hfill \end{array}$$$$\begin{array}{cc}\hfill {D}_{\theta}^{\frac{\beta}{2}}{D}_{1-\theta}^{\frac{\beta}{2}}f\left(x\right)& ={D}_{1}^{\beta}f\left(x\right).\hfill \end{array}$$
- Existence of inverse derivativeFrom (13), the anti-derivative exists when ${\beta}_{2}=-{\beta}_{1}$ and ${\theta}_{1}=-{\theta}_{2}$. Therefore,$${D}_{\theta}^{\beta}{D}_{-\theta}^{-\beta}f\left(x\right)={D}_{-\theta}^{-\beta}{D}_{\theta}^{\beta}f\left(x\right)=f\left(x\right).$$
- Identity operator

- forward derivative, $\theta =\alpha $$${\mathsf{\Psi}}_{\alpha}^{\alpha}\left(\omega \right)={\left|\omega \right|}^{\alpha}{e}^{i\alpha \frac{\pi}{2}\mathrm{sgn}\left(\omega \right)}={\left(i\omega \right)}^{\alpha}$$
- backward derivative, $\theta =-\alpha $$${\mathsf{\Psi}}_{-\alpha}^{\alpha}\left(\omega \right)={\left|\omega \right|}^{\alpha}{e}^{-i\alpha \frac{\pi}{2}\mathrm{sgn}\left(\omega \right)}={(-i\omega )}^{\alpha}$$
- Riesz derivative and potential, $\theta =0$$${\mathsf{\Psi}}_{0}^{\alpha}\left(\omega \right)={\left|\omega \right|}^{\alpha}$$
- Feller derivative and potential, $\theta =1$$${\mathsf{\Psi}}_{1}^{\alpha}\left(\omega \right)=i{\left|\omega \right|}^{\alpha}\mathrm{sgn}\left(\omega \right)$$
- Hilbert transform, $\alpha =0$, $\theta =1$$${\mathsf{\Psi}}_{1}^{0}\left(\omega \right)={e}^{i\frac{\pi}{2}\mathrm{sgn}\left(\omega \right)}=i\mathrm{sgn}\left(\omega \right).$$

- choosing particular values of the parameters $\alpha $ and $\theta $,
- restricting the domain of the function at hand.

## 4. One-Sided Derivatives

#### 4.1. The GL Derivatives

#### 4.2. The Impulse Response

#### 4.3. Liouville’s Derivatives

**Remark**

**2.**

- 1.
- These derivative definitions are equivalent for functions with LT or FT.
- 2.
- For some particular classes of functions this may not be correct, e.g., the L derivatives are better than the LC. Possible causes for this are:
- The convolution makes the functions smoother;
- The derivative may introduce roughness or spikes.

- 3.
- These 2 + 2 formulations lead to most derivatives described in [7,8]. We must reinforce something very important: all the above defined derivatives are valid for functions defined in any interval of $\mathbb{R}$. This means that we do not need to change the definitions to accommodate them to the domain of the function at hand. One thing is the definition, another one is the computation of the derivative. It is a situation similar to the one we find in the LT or FT. We do not need to change the definitions to agree with the domain of the function. Therefore, most derivative definitions in [7,8] have no reason to be considered as autonomous derivatives.
- 4.
- These derivatives do not introduce any initial conditions.
- 5.
- For a given derivative, there is always an anti-derivative.

#### 4.4. RL and C Derivatives

- Riemann–Liouville (RL) derivatives$${}^{RL}{D}_{a+}^{\alpha}f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma}(N-\alpha )}}{\displaystyle \frac{{\mathrm{d}}^{N}}{\mathrm{d}{t}^{N}}}{\int}_{a}^{t}f\left(\tau \right){(t-\tau )}^{N-\alpha -1}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\tau ,\phantom{\rule{1.em}{0ex}}t>a$$$${}^{RL}{D}_{b-}^{\alpha}f\left(t\right)={\displaystyle \frac{{(-1)}^{N}}{\mathrm{\Gamma}(N-\alpha )}}{\displaystyle \frac{{\mathrm{d}}^{N}}{\mathrm{d}{t}^{N}}}{\int}_{t}^{b}f\left(\tau \right){(t-\tau )}^{N-\alpha -1}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\tau ,\phantom{\rule{1.em}{0ex}}t>a.$$
- (Dzherbashian-)Caputo (C) derivatives$${}^{C}{D}_{a+}^{\alpha}f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma}(N-\alpha )}}{\int}_{a}^{t}{f}^{\left(N\right)}\left(\tau \right){(t-\tau )}^{N-\alpha -1}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\tau ,\phantom{\rule{1.em}{0ex}}t>a$$$${}^{C}{D}_{b-}^{\alpha}f\left(t\right)={\displaystyle \frac{{(-1)}^{N}}{\mathrm{\Gamma}(N-\alpha )}}{\int}_{t}^{b}{f}^{\left(N\right)}\left(\tau \right){(t-\tau )}^{N-\alpha -1}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\tau ,\phantom{\rule{1.em}{0ex}}t<b.$$

#### 4.5. Multistep Derivatives

#### 4.6. Second Generation Operators

## 5. Pseudo-Fractional-Derivatives

#### 5.1. Some Comments

#### 5.2. “Derivatives” That Are High-Pass Filters

#### 5.3. “Disguised” Order 1 Derivatives

## 6. Which Derivatives?

- In space problems, we can use the above formulæor the corresponding right-side versions, if there is any privileged direction. If this does not happen, we must use a two-sided derivative, preferably (2).

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Name | Definition | Parameters | Domain | |
---|---|---|---|---|

L | ${D}_{-\infty}^{\alpha}f\left(x\right)=\frac{1}{\mathrm{\Gamma}(N-\alpha )}\frac{{\mathrm{d}}^{N}}{\mathrm{d}{x}^{N}}{\int}_{-\infty}^{x}{(x-\xi )}^{-\alpha +N-1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\gamma =\alpha $ | $\theta =\alpha $ | $x\in \mathbb{R}$ |

${D}_{\infty}^{\alpha}f\left(x\right)=\frac{{(-1)}^{N}}{\mathrm{\Gamma}(N-\alpha )}\frac{{\mathrm{d}}^{N}}{\mathrm{d}{x}^{N}}{\int}_{x}^{+\infty}{(\xi -x)}^{-\alpha +N-1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\gamma =\alpha $ | $\theta =-\alpha $ | $x\in \mathbb{R}$ | |

LC | ${D}_{-\infty}^{\alpha}f\left(x\right)=\frac{1}{\mathrm{\Gamma}(N-\alpha )}{\int}_{-\infty}^{x}{(x-\xi )}^{-\alpha +N-1}\frac{{\mathrm{d}}^{N}f\left(\xi \right)}{\mathrm{d}{\xi}^{N}}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\gamma =\alpha $ | $\theta =\alpha $ | $\phantom{\rule{4pt}{0ex}}x\in \mathbb{R}$ |

${D}_{+\infty}^{\alpha}f\left(x\right)=\frac{{(-1)}^{N}}{\mathrm{\Gamma}(N-\alpha )}{\int}_{x}^{\infty}{(x-\xi )}^{-\alpha +N-1}\frac{{\mathrm{d}}^{N}f\left(\xi \right)}{\mathrm{d}{\xi}^{N}}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\gamma =\alpha $ | $\theta =-\alpha $ | $\phantom{\rule{4pt}{0ex}}x\in \mathbb{R}$ | |

GL | ${D}_{-\infty}^{\alpha}f\left(x\right)=\underset{h\to {0}^{+}}{lim}{h}^{-\alpha}\sum _{k=0}^{\infty}{(-1)}^{k}\left(\begin{array}{c}\alpha \\ k\end{array}\right)f(x-kh)$ | $\gamma =\alpha $ | $\theta =\alpha $ | $x\in \mathbb{R}$ |

${D}_{\infty}^{\alpha}f\left(x\right)=\underset{h\to {0}^{+}}{lim}{h}^{-\alpha}\sum _{k=0}^{\infty}{(-1)}^{k}\left(\begin{array}{c}\alpha \\ k\end{array}\right)f(x+kh)$ | $\gamma =\alpha $ | $\theta =-\alpha $ | $x\in \mathbb{R}$ |

Name | Definition | Parameters | Domain | |
---|---|---|---|---|

RL | ${D}_{{a}^{+}}^{\alpha}f\left(x\right)=\frac{1}{\mathrm{\Gamma}(N-\alpha )}\frac{{\mathrm{d}}^{N}}{\mathrm{d}{x}^{N}}{\int}_{a}^{x}{(x-\xi )}^{-\alpha +N-1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\gamma =\alpha $ | $\theta =\alpha $ | $\phantom{\rule{4pt}{0ex}}x\in [a,b]$ |

${D}_{{b}^{-}}^{\alpha}f\left(x\right)=\frac{{(-1)}^{N}}{\mathrm{\Gamma}(N-\alpha )}\frac{{\mathrm{d}}^{N}}{\mathrm{d}{x}^{N}}{\int}_{x}^{b}{(\xi -x)}^{-\alpha +N-1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\gamma =\alpha $ | $\theta =-\alpha $ | $\phantom{\rule{4pt}{0ex}}x\in [a,b]$ | |

C | ${D}_{{a}^{+}}^{\alpha}f\left(x\right)=\frac{1}{\mathrm{\Gamma}(N-\alpha )}{\int}_{a}^{x}{(x-\xi )}^{-\alpha +N-1}\frac{{\mathrm{d}}^{N}f\left(\xi \right)}{\mathrm{d}{\xi}^{N}}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\gamma =\alpha $ | $\theta =\alpha $ | $x\in [a,b]$ |

${D}_{{b}^{-}}^{\alpha}f\left(x\right)=\frac{{(-1)}^{N}}{\mathrm{\Gamma}(N-\alpha )}{\int}_{x}^{b}{(x-\xi )}^{-\alpha +N-1}\frac{{\mathrm{d}}^{N}f\left(\xi \right)}{\mathrm{d}{\xi}^{N}}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\gamma =\alpha $ | $\theta =-\alpha $ | $x\in [a,b]$ | |

GL | $D}_{{a}^{+}}^{\alpha}f\left(x\right)=\underset{h\to {0}^{+}}{lim}\frac{{\displaystyle \sum _{k=0}^{\u230a\frac{x-a}{h}\u230b}{(-1)}^{k}\left(\begin{array}{c}\alpha \\ k\end{array}\right)f(x-kh)}}{{h}^{\alpha}$ | $\gamma =\alpha $ | $\theta =\alpha $ | $x\in [a,b]$ |

$D}_{{a}^{-}}^{\alpha}f\left(x\right)=\underset{h\to {0}^{+}}{lim}\frac{{\displaystyle \sum _{k=0}^{\u230a\frac{a-x}{h}\u230b}{(-1)}^{k}\left(\begin{array}{c}\alpha \\ k\end{array}\right)f(x+kh)}}{{h}^{\alpha}$ | $\gamma =\alpha $ | $\theta =-\alpha $ | $x\in [a,b]$ |

Name | Definition | Domain |
---|---|---|

Hadamard | ${D}_{{0}^{+}}^{\alpha}f\left(x\right)=\frac{1}{\mathrm{\Gamma}(-\alpha )}{\int}_{0}^{x}\xi {\left(log\frac{\xi}{x}\right)}^{-\alpha -1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\alpha <0,\phantom{\rule{4pt}{0ex}}x\ge 0$ |

${D}_{{0}^{+}}^{\alpha}f\left(x\right)=\frac{1}{\mathrm{\Gamma}(1-\alpha )}{\int}_{0}^{x}\xi {\left(log\frac{x}{\xi}\right)}^{-\alpha -1}\left(f\left(x\right)-f\left(\xi \right)\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\alpha >0,\phantom{\rule{4pt}{0ex}}x\ge 0$ | |

Quantum | $D}_{q}^{\alpha}f\left(x\right)={x}^{-\alpha}\underset{q\to 1}{lim}\frac{{\displaystyle \sum _{j=0}^{+\infty}{\left[\begin{array}{c}\alpha \\ j\end{array}\right]}_{q}{(-1)}^{j}{q}^{\frac{j(j+1)}{2}}{q}^{-j\alpha}f\left({q}^{j}x\right)}}{{(1-q)}^{\alpha}$ | |

$D}_{{q}^{-1}}^{\alpha}f\left(x\right)={x}^{-\alpha}\underset{q\to 1}{lim}\frac{{\displaystyle \sum _{j=0}^{+\infty}{\left[\begin{array}{c}\alpha \\ j\end{array}\right]}_{q}{(-1)}^{j}{q}^{\frac{j(j-1)}{2}}{q}^{-j\alpha}f\left({q}^{-j}x\right)}}{{\left(1-{q}^{-1}\right)}^{\alpha}$ | ||

$\left[\begin{array}{c}\alpha \\ j\end{array}\right]}_{q}=\frac{{\displaystyle \prod _{i=0}^{j-1}\left(1-{q}^{\alpha +i}\right)}}{{\displaystyle \frac{1-{q}^{j}}{1-q}}$ | ||

Marchaud | ${D}_{{0}^{+}}^{\alpha}f\left(x\right)=\frac{\alpha}{\mathrm{\Gamma}(1-\alpha )}{\int}_{0}^{+\infty}{\xi}^{-\alpha -1}\left(f\left(x\right)-f(x-\xi )\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\alpha >0$ |

${D}_{{0}^{-}}^{\alpha}f\left(x\right)=\frac{\alpha}{\mathrm{\Gamma}(1-\alpha )}{\int}_{0}^{+\infty}{\xi}^{-\alpha -1}\left(f\left(x\right)-f(x+\xi )\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\alpha <0$ | |

Jumarie | ${D}_{{0}^{+}}^{\alpha}f\left(x\right)=\frac{1}{\mathrm{\Gamma}(N-\alpha )}\frac{{\mathrm{d}}^{N}}{\mathrm{d}{x}^{N}}{\int}_{0}^{x}{(x-\xi )}^{-\alpha +N-1}\left(f\left(\xi \right)-f\left(0\right)\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\alpha >0,\phantom{\rule{4pt}{0ex}}x\ge 0$ |

Erdélyi-Kober | ${D}_{{a}^{+},\sigma ,\eta}^{\alpha}f\left(x\right)=\frac{\sigma {x}^{\sigma (\eta -\alpha )}}{\mathrm{\Gamma}(-\alpha )}{\int}_{a}^{x}{({x}^{\sigma}-{\xi}^{\sigma})}^{-\alpha -1}{\xi}^{\sigma (1+\eta )-1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\alpha <0,\phantom{\rule{4pt}{0ex}}x\ge a$ |

${D}_{{a}^{-},\sigma ,\eta}^{\alpha}f\left(x\right)=\frac{\sigma {x}^{-\sigma \alpha}}{\mathrm{\Gamma}(-\alpha )}{\int}_{x}^{a}{({\xi}^{\sigma}-{x}^{\sigma})}^{-\alpha -1}{\xi}^{\sigma (1+\alpha -\eta )-1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\alpha <0,\phantom{\rule{4pt}{0ex}}x\le a$ | |

${D}_{{a}^{+},\sigma ,\eta}^{\alpha}f\left(x\right)={x}^{-\sigma (\eta +\alpha )}{\left(\frac{1}{\sigma {x}^{\sigma -1}}\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{N}{x}^{\sigma (\alpha +N+\eta )}{D}_{{a}^{+},\sigma ,\eta}^{-N-\alpha}f\left(x\right)$ | $\alpha >0,\phantom{\rule{4pt}{0ex}}x\ge a$ | |

${D}_{{a}^{-},\sigma ,\eta}^{\alpha}f\left(x\right)={x}^{\sigma \eta}{\left(\frac{1}{\sigma {x}^{\sigma -1}}\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{N}{x}^{\sigma (N-\eta )}{D}_{{a}^{-},\sigma ,\eta}^{-N-\alpha}f\left(x\right)$ | $\alpha >0,\phantom{\rule{4pt}{0ex}}x\le a$ | |

k-Hilfer | ${D}_{k,{a}^{+}}^{\alpha}f\left(x\right)=\frac{1}{{k}^{-\frac{\alpha}{k}}\mathrm{\Gamma}(-\alpha )}{\int}_{a}^{x}{(x-\xi )}^{-\frac{\alpha}{k}-1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\alpha <0,\phantom{\rule{4pt}{0ex}}x\ge a$ |

${D}_{k,{a}^{+}}^{\mu ,\nu}f\left(x\right)={D}_{k,{a}^{+}}^{\nu (\mu -1)}\frac{\mathrm{d}}{\mathrm{d}x}{D}_{k,{a}^{+}}^{(1-\nu )(\mu -1)}f\left(x\right)$ | $\begin{array}{c}0\le \mu \le 1,\phantom{\rule{4pt}{0ex}}0<\nu <1,\\ x\ge a\end{array}$ | |

Hilfer-Katugampola | ${D}_{\rho ,{a}^{+}}^{\alpha}f\left(x\right)=\frac{{\rho}^{1+\alpha}}{\mathrm{\Gamma}(-\alpha )}{\int}_{a}^{x}{({x}^{\rho}-{\xi}^{\rho})}^{-\alpha -1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\rho >0,\phantom{\rule{4pt}{0ex}}\alpha <0,\phantom{\rule{4pt}{0ex}}x\ge a$ |

${D}_{\rho ,{a}^{-}}^{\alpha}f\left(x\right)=\frac{{\rho}^{1+\alpha}}{\mathrm{\Gamma}(-\alpha )}{\int}_{x}^{a}{({x}^{\rho}-{\xi}^{\rho})}^{-\alpha -1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\rho >0,\phantom{\rule{4pt}{0ex}}\alpha <0,\phantom{\rule{4pt}{0ex}}x\le a$ | |

${D}_{\rho ,{a}^{\pm}}^{\alpha ,\beta}f\left(x\right)=\pm {D}_{\rho ,{a}^{\pm}}^{\beta (\alpha -1)}\left({x}^{1-\rho}\frac{\mathrm{d}}{\mathrm{d}x}{D}_{\rho ,{a}^{\pm}}^{(1-\beta )(\alpha -1)}f\left(x\right)\right)$ | $\begin{array}{c}\rho >0,\phantom{\rule{4pt}{0ex}}1>\alpha >0,\\ 0\ge \beta \ge 0\end{array}$ | |

$\psi $-Hilfer, with | ${D}_{{a}^{+}}^{\alpha ,\psi \left(x\right)}f\left(x\right)=\frac{1}{\mathrm{\Gamma}(-\alpha )}{\int}_{a}^{x}{\psi}^{\prime}\left(x\right){(\psi \left(x\right)-\psi \left(\xi \right))}^{-\alpha -1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\alpha <0,\phantom{\rule{4pt}{0ex}}I=[a,x]$ |

$\psi \left(x\right)\in {C}^{N}(I,\mathbb{R})$ | ${D}_{{a}^{-}}^{\alpha ,\psi \left(x\right)}f\left(x\right)=\frac{1}{\mathrm{\Gamma}(-\alpha )}{\int}_{x}^{a}{\psi}^{\prime}\left(x\right){(\psi \left(\xi \right)-\psi \left(x\right))}^{-\alpha -1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $\alpha <0,\phantom{\rule{4pt}{0ex}}I=[x,a]$ |

${\psi}^{\prime}\left(x\right)\ne 0,\phantom{\rule{4pt}{0ex}}\forall x\in I$ | ${D}_{{a}^{+}}^{\alpha ,\beta ,\psi \left(x\right)}f\left(x\right)={D}_{{a}^{+}}^{\beta (\alpha -N),\psi \left(x\right)}{\left(\frac{1}{{\psi}^{\prime}\left(x\right)}\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{N}{D}_{{a}^{+}}^{(1-\beta )(\alpha -N),\psi \left(x\right)}f\left(x\right)$ | $\begin{array}{c}\alpha >0,\phantom{\rule{4pt}{0ex}}0\le \beta \le 1,\\ I=[a,x]\end{array}$ |

$\begin{array}{c}{x}_{2}>{x}_{1}\Rightarrow \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\\ \Rightarrow \psi \left({x}_{2}\right)>\psi \left({x}_{1}\right)\end{array}$ | ${D}_{{a}^{-}}^{\alpha ,\beta ,\psi \left(x\right)}f\left(x\right)={D}_{{a}^{-}}^{\beta (\alpha -N),\psi \left(x\right)}{\left(\frac{1}{{\psi}^{\prime}\left(x\right)}\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{N}{D}_{{a}^{+}}^{(1-\beta )(\alpha -N),\psi \left(x\right)}f\left(x\right)$ | $\begin{array}{c}\alpha >0,\phantom{\rule{4pt}{0ex}}0\le \beta \le 1,\\ I=[x,a]\end{array}$ |

**Table 4.**“Derivatives” with non-singular kernel; in all cases, the order verifies $0<\alpha <1$, and $M\left(\alpha \right)$ verifies $M\left(0\right)=M\left(1\right)=1$.

Name | Definition | Domain |
---|---|---|

Caputo-Fabrizio | ${D}_{{a}^{+}}^{\alpha}f\left(x\right)=\frac{M\left(\alpha \right)}{1-\alpha}{\int}_{a}^{x}{e}^{-\frac{\alpha (x-\xi )}{1-\alpha}}\frac{\mathrm{d}f\left(\xi \right)}{\mathrm{d}\xi}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $x\ge a$ |

Yang et al. | ${D}_{{a}^{+}}^{\alpha}f\left(x\right)=\frac{M\left(\alpha \right)}{1-\alpha}\frac{\mathrm{d}}{\mathrm{d}x}{\int}_{a}^{x}{e}^{-\frac{\alpha (x-\xi )}{1-\alpha}}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $x\ge a$ |

Atangana-Baleanu-Caputo | ${D}_{{a}^{+}}^{\alpha}f\left(x\right)=\frac{M\left(\alpha \right)}{1-\alpha}{\int}_{a}^{x}{E}_{\alpha}\left(-\frac{\alpha {(x-\xi )}^{\alpha}}{1-\alpha}\right)\frac{\mathrm{d}f\left(\xi \right)}{\mathrm{d}\xi}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $x\ge a$ |

Atangana-Baleanu-Riemann–Liouville | ${D}_{{a}^{+}}^{\alpha}f\left(x\right)=\frac{M\left(\alpha \right)}{1-\alpha}\frac{\mathrm{d}}{\mathrm{d}x}{\int}_{a}^{x}{E}_{\alpha}\left(-\frac{\alpha {(x-\xi )}^{\alpha}}{1-\alpha}\right)f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $x\ge a$ |

Generalized Caputo | ${D}_{{a}^{+}}^{\alpha ,\beta}f\left(x\right)=\frac{M\left(\alpha \right)}{1-\alpha}{\int}_{a}^{x}{E}_{\beta}\left(-\frac{\alpha {(x-\xi )}^{\beta}}{1-\alpha}\right)\frac{\mathrm{d}f\left(\xi \right)}{\mathrm{d}\xi}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $0\le \beta \le 1,\phantom{\rule{4pt}{0ex}}x\ge a$ |

Generalized Riemann–Liouville | ${D}_{{a}^{+}}^{\alpha ,\beta}f\left(x\right)=\frac{M\left(\alpha \right)}{1-\alpha}\frac{\mathrm{d}}{\mathrm{d}x}{\int}_{a}^{x}{E}_{\beta}\left(-\frac{\alpha {(x-\xi )}^{\beta}}{1-\alpha}\right)f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $0\le \beta \le 1,\phantom{\rule{4pt}{0ex}}x\ge a$ |

Caputo-Fabrizio, Gaussian kernel | ${D}_{{a}^{+}}^{\alpha}f\left(x\right)=\frac{1+{\alpha}^{2}}{\sqrt{{\pi}^{\alpha}(1-\alpha )}}{\int}_{a}^{x}{e}^{-\frac{\alpha {(x-\xi )}^{2}}{1-\alpha}}\frac{\mathrm{d}f\left(\xi \right)}{\mathrm{d}\xi}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $f\left(a\right)=0,\phantom{\rule{4pt}{0ex}}x\ge a$ |

Sun-Hao-Zhang-Baleanu | ${D}_{{a}^{+}}^{\alpha}f\left(x\right)=\frac{M\left(\alpha \right)}{{(1-\alpha )}^{\frac{1}{\alpha}}}{\int}_{a}^{x}{e}^{-\frac{\alpha {(x-\xi )}^{\alpha}}{1-\alpha}}\frac{\mathrm{d}f\left(\xi \right)}{\mathrm{d}\xi}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi$ | $x\ge a$ |

Name | Definition | Domain |
---|---|---|

Kolwankar | ${D}^{\alpha}f\left(x\right)=\underset{\xi \to x}{lim}{D}_{x}^{\alpha}\left(f\left(\xi \right)-f\left(x\right)\right)$ where ${D}_{x}^{\alpha}$ is the RL derivative | $x\in {\mathbb{R}}^{+}$ |

Chen | $D}^{\alpha}f\left(x\right)=\underset{\xi \to x}{lim}\frac{f\left(x\right)-f\left(\xi \right)}{{x}^{\alpha}-{\xi}^{\alpha}$ | $x\in {\mathbb{R}}^{+}$ |

Conformable | $D}^{\alpha}f\left(x\right)=\underset{\epsilon \to 0}{lim}\frac{f(x+\epsilon {x}^{1-\alpha})-f\left(x\right)}{\epsilon$ | $x\in {\mathbb{R}}^{+}$ |

Katugampola | $D}^{\alpha}f\left(x\right)=\underset{\epsilon \to 0}{lim}\frac{f\left(x\phantom{\rule{0.166667em}{0ex}}{e}^{\epsilon {x}^{-\alpha}}\right)-f\left(x\right)}{\epsilon$ | $x\in {\mathbb{R}}^{+}$ |

M | $D}^{\alpha ,\beta}f\left(x\right)=\underset{\epsilon \to 0}{lim}\frac{f\left(x\phantom{\rule{0.166667em}{0ex}}{E}_{\beta}\left(\epsilon {x}^{-\alpha}\right)\right)-f\left(x\right)}{\epsilon$ | $x\in {\mathbb{R}}^{+},\beta >0$ |

Deformable | $D}^{\alpha}f\left(x\right)=\underset{\epsilon \to 0}{lim}\frac{(1+\epsilon -\epsilon \alpha )f(x+\epsilon \alpha )-f\left(x\right)}{\epsilon$ | $x\in {\mathbb{R}}^{+}$ |

Beta | $D}^{\alpha}f\left(x\right)=\underset{\epsilon \to 0}{lim}\frac{f\left(x+\epsilon {\left(x+\frac{1}{\mathrm{\Gamma}\left(\alpha \right)}\right)}^{1-\alpha}\right)-f\left(x\right)}{\epsilon$ | $0<\alpha \le 1,\phantom{\rule{4pt}{0ex}}x\in {\mathbb{R}}^{+}$ |

AGO | $D}^{\alpha}f\left(x\right)=\underset{\epsilon \to 0}{lim}\frac{f\left(x+\u03f5{\left(\psi \left(x\right)\right)}^{1-\alpha}\right)-f\left(x\right)}{\epsilon$ | $0<\alpha <1,x\in {\mathbb{R}}^{+}$ |

$\phantom{\rule{1.em}{0ex}}\psi \left(x\right)\in C\left({\mathbb{R}}^{+}\right),\phantom{\rule{4pt}{0ex}}{\psi}^{\prime}\left(x\right)\ne 0$ | ||

Generalized | $D}^{\alpha}f\left(x\right)=\underset{\epsilon \to 0}{lim}\frac{f\left(x-\psi \left(x\right)+\psi \left(x\right)\phantom{\rule{0.166667em}{0ex}}{e}^{\frac{\u03f5{\left(\psi \left(x\right)\right)}^{-\alpha}}{{\psi}^{\prime}\left(x\right)}}\right)-f\left(x\right)}{\epsilon$ | $0<\alpha <1,\phantom{\rule{4pt}{0ex}}x\in {\mathbb{R}}^{+}$ |

$\phantom{\rule{1.em}{0ex}}\psi \left(x\right)\in C\left({\mathbb{R}}^{+}\right),\phantom{\rule{4pt}{0ex}}{\psi}^{\prime}\left(x\right)\ne 0$ | ||

General conformable | $D}_{\psi}^{\alpha}f\left(x\right)=\underset{\epsilon \to 0}{lim}\frac{f(x+\epsilon \phantom{\rule{0.166667em}{0ex}}\psi (x,\alpha \left)\right)-f\left(x\right)}{\epsilon$ | $\alpha >0,\phantom{\rule{4pt}{0ex}}x\in {\mathbb{R}}^{+}$ |

Lazopoulos’s Lambda | $D}_{\Lambda}^{\alpha}f\left(x\right)=\frac{{\displaystyle \frac{1}{\mathrm{\Gamma}(N-\alpha )}\frac{{\mathrm{d}}^{N}}{\mathrm{d}{x}^{N}}{\int}_{a}^{x}{(x-\xi )}^{-\alpha +N-1}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi}}{{\displaystyle \frac{1}{\mathrm{\Gamma}(N-\alpha )}\frac{{\mathrm{d}}^{N}}{\mathrm{d}{x}^{N}}{\int}_{a}^{x}{(x-\xi )}^{-\alpha +N-1}\xi \phantom{\rule{0.166667em}{0ex}}\mathrm{d}\xi}$ | $\phantom{\rule{4pt}{0ex}}x\in {\mathbb{R}}^{+}$ |

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**MDPI and ACS Style**

Valério, D.; Ortigueira, M.D.; Lopes, A.M.
How Many Fractional Derivatives Are There? *Mathematics* **2022**, *10*, 737.
https://doi.org/10.3390/math10050737

**AMA Style**

Valério D, Ortigueira MD, Lopes AM.
How Many Fractional Derivatives Are There? *Mathematics*. 2022; 10(5):737.
https://doi.org/10.3390/math10050737

**Chicago/Turabian Style**

Valério, Duarte, Manuel D. Ortigueira, and António M. Lopes.
2022. "How Many Fractional Derivatives Are There?" *Mathematics* 10, no. 5: 737.
https://doi.org/10.3390/math10050737