# Which Derivative?

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## Abstract

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## 1. Introduction

#### Remarks

- We work on $\mathbf{R}$.
- We use the two-sided Laplace transform (LT):$$F\left(s\right)=\underset{\mathbf{R}}{\int}f\left(t\right){e}^{-st}\mathrm{d}t$$
- The Fourier transform (FT) is obtained from the LT through the substitution $s=i\omega $ with $\omega \in \mathbf{R}$
- The functions and distributions have Laplace and/or Fourier transforms
- Current properties of the Dirac delta distribution and its derivatives will be used
- We will work with the usual convolution$$f\left(t\right)*g\left(t\right)=\underset{\mathbf{R}}{\int}f\left(\tau \right)g(t-\tau )\mathrm{d}\tau $$
- The fractional derivative order is assumed to be any real number
- The multi-valued expressions ${s}^{\alpha}$ and ${(-s)}^{\alpha}$ will frequently be used. To obtain functions from them we will fix for branch-cut lines the negative real half axis for the first and the positive real half axis for the second; for both the first Riemann surface is chosen.

## 2. Some Classic Results

#### 2.1. The Derivative Operators and Their Inverses

- The different time flow shows its influence: the causality (anti-causality) is clearly stated,
- We have ${D}_{f}^{-1}{D}_{f}f\left(t\right)={D}_{f}{D}_{f}^{-1}f\left(t\right)=f\left(t\right)$ and ${D}_{b}^{-1}{D}_{b}f\left(t\right)={D}_{b}{D}_{b}^{-1}f\left(t\right)=f\left(t\right)$. We will call ${D}^{-1}$ “anti-derivative” [6].

- In the derivative case ($+N$) the summation goes only to N, since the ${(-N)}_{n}$ becomes null for $n>N$,
- Let ${n}_{1}$ and ${n}_{2}$ be two integer values. With (7) and under the assumed functional space we can write$${D}_{f}^{{n}_{1}}{D}_{f}^{{n}_{2}}f\left(t\right)={D}_{f}^{{n}_{2}}{D}_{f}^{{n}_{1}}f\left(t\right)={D}_{f}^{{n}_{1}+{n}_{2}}f\left(t\right)$$For the backward derivatives the situation is similar. This result is straightforward using the properties of the binomial coefficients or the $\mathcal{Z}$ transform [15]

- The constant function$$c\left(t\right)=1,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{2.em}{0ex}}t\in \mathbf{R},$$
- The Heaviside, or unit step, function$$u\left(t\right)=\left\{\begin{array}{c}1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t>0\hfill \\ 0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t<0\hfill \end{array}\right.,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{2.em}{0ex}}t\in \mathbf{R}.$$

#### 2.2. System Interpretation

- The terms with negative exponents represent two TF corresponding to two disjoint regions of convergence, namely $\mathbf{R}e\left(s\right)>0$ (causal system) and $\mathbf{R}e\left(s\right)<0$ (anti-causal system),
- The terms with positive or null exponents are analytic on the whole complex plane and consequently there is no causality involved.

#### 2.3. Other Important Results

- Derivative eigenfunctionsReturning back to (18)$${D}^{n}{e}^{zt}={z}^{n}{e}^{zt},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\in \mathbf{Z},\phantom{\rule{0.277778em}{0ex}}t\in \mathbf{R},\phantom{\rule{0.277778em}{0ex}}z\in \mathbf{C}.$$This result, for the particular case of $z=i\omega ,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\omega \in {\mathbf{R}}^{+}$, $i=\sqrt{-1}$, yields$$\left\{\begin{array}{c}{D}^{n}cos\left(\omega t\right)=\omega cos(\omega t+n\xb7\frac{\pi}{2})\hfill \\ {D}^{n}sin\left(\omega t\right)=\omega sin(\omega t+n\xb7\frac{\pi}{2})\hfill \end{array}\right.,\phantom{\rule{0.277778em}{0ex}}t\in \mathbf{R}.$$These results are also valid for negative n.
- The Leibniz relation for the productThe classic Leibniz relation gives the derivative of the product of two functions and can be written as$${D}_{f}^{n}\left[f\left(t\right)g\left(t\right)\right]=\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right){D}_{f}^{k}f\left(t\right){D}_{f}^{n-k}g\left(t\right),$$

## 3. Backward Compatibility in Fractional Calculus

#### 3.1. Some Considerations

#### 3.2. Causal FC Based on the Incremental Ratio

- Linearity
- Additivity and Commutativity of the orders. If we apply (22) twice for any two orders, we have$${D}_{f}^{\alpha}{D}_{f}^{\beta}f\left(t\right)={D}_{f}^{\beta}{D}_{f}^{\alpha}f\left(t\right)={D}_{f}^{\alpha +\beta}f\left(t\right).$$
- Neutral element$${D}_{f}^{\alpha}{D}_{f}^{-\alpha}f\left(t\right)={D}_{f}^{0}f\left(t\right)=f\left(t\right).$$From (25) we conclude that there is always an inverse element, that is, for every $\alpha $ there is always the $-\alpha $ order derivative.
- Backward compatibility ($n\in \mathbf{N}$)If $\alpha =n$, then:$${D}_{f}^{n}f\left(t\right)=\underset{h\to 0}{lim}\frac{{\sum}_{k=0}^{n}{(-1)}^{k}\left(\genfrac{}{}{0pt}{}{n}{k}\right)f(t-kh)}{{h}^{n}}$$We obtain this expression repeating the first order derivative.If $\alpha =-n$, then:$${D}_{f}^{-n}f\left(t\right)=\underset{h\to 0}{lim}\sum _{k=0}^{n}\frac{{\left(n\right)}_{k}}{k!}f(t-kh)\xb7{h}^{n},$$
- We can apply the two-sided LT to (22) and (23) to obtain$$\mathcal{L}\left[{D}_{f}^{\alpha}f\left(t\right)\right]={s}^{\alpha}\mathcal{L}\left[f\left(t\right)\right],$$
- The generalized Leibniz rule for the productThe generalized Leibniz rule gives the FD of the product of two functions. It is one of the most important characteristics of the FD [18,19] and assumes the format [1]$${D}_{f}^{\alpha}\left[f\left(t\right)g\left(t\right)\right]=\sum _{k=0}^{\infty}\left(\genfrac{}{}{0pt}{}{\alpha}{k}\right){D}_{f}^{k}f\left(t\right){D}_{f}^{\alpha -k}g\left(t\right).$$For the backward the formula is identical.

#### 3.3. Some Examples

- Constant functionIf $f\left(t\right)=1$, for every $t\in \mathbf{R}$ and $\alpha \in \mathbf{R}$, then we have$$\begin{array}{c}\hfill {D}^{\alpha}f\left(t\right)=\underset{h\to {0}^{+}}{lim}\frac{{\sum}_{k=0}^{\infty}\left(\begin{array}{c}\alpha \\ k\end{array}\right){(-1)}^{k}}{{h}^{\alpha}}=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\text{}\alpha 0\hfill \\ \infty ,\hfill & \mathrm{if}\text{}\alpha 0\hfill \end{array}\right..\end{array}$$
- Causal power functionWe calculate the FD of the Heaviside function. Starting from [1]:$$\sum _{k=0}^{n}\left(\begin{array}{c}\alpha \\ k\end{array}\right){(-1)}^{k}=\left(\begin{array}{c}\alpha -1\\ n\end{array}\right){(-1)}^{n}=\frac{1}{\mathsf{\Gamma}(1-\alpha )}\frac{\mathsf{\Gamma}(-\alpha +n+1)}{\mathsf{\Gamma}(n+1)},$$$${D}^{\alpha}u\left(t\right)=\frac{{t}^{-\alpha}}{\mathsf{\Gamma}(1-\alpha )}u\left(t\right).$$As $\delta \left(t\right)=Du\left(t\right)$, we deduce using (24) that$${D}^{\alpha}\delta \left(t\right)=\frac{{t}^{-\alpha -1}}{\mathsf{\Gamma}(\alpha +1)}u\left(t\right).$$With (30) it is possible to obtain the derivative of any order of the continuous function $p\left(t\right)={t}^{\beta}u\left(t\right)$, with $\beta >0$. The $LT$ of $p\left(t\right)$, yields $P\left(s\right)=\frac{\mathsf{\Gamma}(\beta +1)}{{s}^{\beta +1}}$, for $\mathbf{R}e\left(s\right)>0$ and the FD of order $\alpha $ is given by ${s}^{\alpha}\frac{\mathsf{\Gamma}(\beta +1)}{{s}^{\beta +1}}$. Therefore, the expression$${D}^{\alpha}{t}^{\beta}u\left(t\right)=\frac{\mathsf{\Gamma}(\beta +1)}{\mathsf{\Gamma}(\beta -\alpha +1)}{t}^{\beta -\alpha}u\left(t\right),$$$${D}^{\alpha}\frac{{t}^{\alpha -1}}{\mathsf{\Gamma}(\alpha +1)}u\left(t\right)=\frac{{t}^{-1}}{\mathsf{\Gamma}\left(0\right)}u\left(t\right)=\delta \left(t\right).$$

#### 3.4. Obtaining Integral Formulations

- ${D}_{f}^{N}\left[g\left(t\right)\right]$ has a worst analytic behaviour than $g\left(t\right)$; eventually it can be discontinuous.
- The convolution has a smoothing effect. Therefore, in the left side above we are computing the derivative of a function with “better behaviour”.

- If $f\left(t\right)$ has Laplace transform with a nondegenerate region of convergence, the three derivatives give the same result,
- The Liouville-Caputo derivative demands too much from analytical point of view, since it needs the unnecessary existence of the ${N}^{th}$ order derivative,
- If $f\left(t\right)=1,\phantom{\rule{0.277778em}{0ex}}t\in \mathbf{R}$ the Riemann-Liouville derivative does not exist, since the integral is divergent.

#### 3.5. The TF of the Differintegrator

#### 3.6. Classic Riemann-Liouville and Caputo Derivatives

## 4. The Linear Differential Equations

#### 4.1. The Transfer Function and the Impulse Response

#### 4.2. The Initial Condition Problem

- Equation (45) is defined for any $t\in \mathbf{R}$,
- Our observation window is the unit step $u(t-{t}_{0})$,
- The IC depend on the structure of the system and are independent of the tools that we adopt for the analysis,
- The IC are the values assumed by the variables at the instant of opening the observation window.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

C | Caputo |

FT | Fourier transform |

FD | Fractional derivative |

FI | Fractional integral |

GL | Grünwald-Letnikov |

IC | Initial conditions |

L | Liouville |

LT | Laplace transform |

RL | Riemann-Liouville |

TF | Transfer function |

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**MDPI and ACS Style**

Ortigueira, M.; Machado, J.
Which Derivative? *Fractal Fract.* **2017**, *1*, 3.
https://doi.org/10.3390/fractalfract1010003

**AMA Style**

Ortigueira M, Machado J.
Which Derivative? *Fractal and Fractional*. 2017; 1(1):3.
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**Chicago/Turabian Style**

Ortigueira, Manuel, and José Machado.
2017. "Which Derivative?" *Fractal and Fractional* 1, no. 1: 3.
https://doi.org/10.3390/fractalfract1010003