# Non-Debye Relaxations: The Ups and Downs of the Stretched Exponential vs. Mittag–Leffler’s Matchings

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## Abstract

**:**

## 1. Introduction

## 2. Characteristic Exponents and Stochastic Description—A Brief Tutorial

## 3. Spectral Function for the KWW Pattern

## 4. Comparison of Characteristic Exponents

- (a)
- The characteristic exponents for CC relaxation model are given by the power-law functions$${\widehat{\mathsf{\Psi}}}_{CC}\left(s\right)={\left(s\tau \right)}^{\alpha}\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}{\widehat{\mathsf{\Phi}}}_{CC}\left(s\right)={\left(s\tau \right)}^{1-\alpha}$$$${\widehat{\mathsf{\Psi}}}_{A;KWW}\left(s\right)\sim {\left(s\tau \right)}^{\alpha}/\Gamma (1+\alpha )\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{\widehat{\mathsf{\Phi}}}_{A;KWW}\left(s\right)\sim \Gamma (1+\alpha ){\left(s\tau \right)}^{1-\alpha},\phantom{\rule{1.em}{0ex}}s\tau \gg 1$$We should also observe that ${\widehat{\mathsf{\Psi}}}_{CC}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{CC}\left(s\right)$, as well as ${\widehat{\mathsf{\Psi}}}_{KWW}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{KWW}\left(s\right)$ are CBFs and by construction satisfy the Sonine condition.
- (b)
- In case of the HN relaxation we have$${\widehat{\mathsf{\Psi}}}_{HN}\left(s\right)=\{{[1+{\left(s\tau \right)}^{\alpha}]}^{\beta}-1\}\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}{\widehat{\mathsf{\Phi}}}_{HN}\left(s\right)=s{\{{[1+{\left(s\tau \right)}^{\alpha}]}^{\beta}-1\}}^{-1}.$$For large s the leading asymptotic term of ${\widehat{\mathsf{\Psi}}}_{HN}\left(s\right)$ is ${\left(s\tau \right)}^{\alpha \beta}$. For small s the relevant asymptotics is got if we rewrite ${\widehat{\mathsf{\Psi}}}_{HN}\left(s\right)$ as the series ${\sum}_{r\ge 0}\Gamma (1+\beta ){\left(s\tau \right)}^{\alpha r}/[r!\Gamma (1+\beta -r)]-1$ whose first two terms (i.e., the terms with $r=0$ and $r=1$) give the asymptotics of ${\widehat{\mathsf{\Psi}}}_{HN}\left(s\right)$ proportional to $\beta {\left(s\tau \right)}^{\alpha}$. Gathered together the asymptotic behavior of ${\widehat{\mathsf{\Psi}}}_{HN}\left(s\right)$ reads$${\widehat{\mathsf{\Psi}}}_{A;HN}\left(s\right)\sim {\left(s\tau \right)}^{\alpha \beta},\phantom{\rule{1.em}{0ex}}s\tau \gg 1,\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}{\widehat{\mathsf{\Psi}}}_{A;HN}\left(s\right)\sim \beta {\left(s\tau \right)}^{\alpha},\phantom{\rule{1.em}{0ex}}s\tau \ll 1$$$${\widehat{\mathsf{\Phi}}}_{A;HN}\left(s\right)\sim {\left(s\tau \right)}^{1-\alpha}/\beta ,\phantom{\rule{1.em}{0ex}}s\tau \gg 1,\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{\widehat{\mathsf{\Phi}}}_{A;HN}\left(s\right)\sim {\left(s\tau \right)}^{1-\alpha \beta},\phantom{\rule{1.em}{0ex}}s\tau \ll 1.$$As in the previous example also here ${\widehat{\mathsf{\Psi}}}_{A;HN}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{A;HN}\left(s\right)$ are CBFs for $\alpha ,\beta \in (0,1]$. The power-law asymptotics given by Equation (17) for $s\tau \gg 1$ shows that in order to match ${\widehat{\mathsf{\Psi}}}_{KWW}\left(s\right)$ the relations of exponentials ${\alpha}_{HN}{\beta}_{HN}={\alpha}_{KWW}$ has to be satisfied. It means that ${\alpha}_{HN}$ may be chosen arbitrarily if simultaneously ${\beta}_{HN}={\alpha}_{KWW}/{\alpha}_{HN}$. Thus, the small s asymptotics of ${\widehat{\mathsf{\Psi}}}_{A;HN}\left(s\right)$ becomes incompatible with the asymptotics of ${\widehat{\mathsf{\Psi}}}_{A;KWW}\left(s\right)$ and matches it only for $\beta =1$ which is the condition reducing the HN pattern to the CC one. Figure 2, with ${\alpha}_{KWW}=1/3$, ${\alpha}_{HN}=5/6$, and ${\beta}_{HN}=2/5$, shows that for large $\tau s$${\widehat{\mathsf{\Psi}}}_{HN}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{HN}\left(s\right)$ fit well ${\widehat{\mathsf{\Psi}}}_{KWW}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{KWW}\left(s\right)$, respectively, but the matching breaks down for small $s\tau $.
- (c)
- The characteristic exponents of the JWS model are equal to$${\widehat{\mathsf{\Psi}}}_{JWS}\left(s\right)={\{{[1+{\left(s\tau \right)}^{-\alpha}]}^{\beta}-1\}}^{-1}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{\widehat{\mathsf{\Phi}}}_{JWS}\left(s\right)=s\{{[1+{\left(s\tau \right)}^{-\alpha}]}^{\beta}-1\}$$Their asymptotics read$$\begin{array}{cc}\hfill {\widehat{\mathsf{\Psi}}}_{A;JWS}\left(s\right)\sim {s}^{\alpha}/\beta ,\phantom{\rule{1.em}{0ex}}s\tau \gg 1,\phantom{\rule{2.em}{0ex}}& \mathrm{and}\phantom{\rule{2.em}{0ex}}{\widehat{\mathsf{\Psi}}}_{A;JWS}\left(s\right)\sim {\left(s\tau \right)}^{\alpha \beta},\phantom{\rule{1.em}{0ex}}s\tau \ll 1,\hfill \end{array}$$$$\begin{array}{cc}\hfill {\widehat{\mathsf{\Phi}}}_{A;JWS}\left(s\right)\sim {s}^{1-\alpha \beta},\phantom{\rule{1.em}{0ex}}s\tau \gg 1,\phantom{\rule{2.em}{0ex}}& \mathrm{and}\phantom{\rule{2.em}{0ex}}{\widehat{\mathsf{\Phi}}}_{A;JWS}\left(s\right)\sim \beta {\left(s\tau \right)}^{1-\alpha},\phantom{\rule{1.em}{0ex}}s\tau \ll 1.\hfill \end{array}$$As in the previous case also here the asymptotics’ presented by Equation (20) are given by CBFs for $\alpha ,\beta \in (0,1]$. The comparison between ${\widehat{\mathsf{\Psi}}}_{KWW}\left(s\right)$ and ${\widehat{\mathsf{\Psi}}}_{JWS}\left(s\right)$, as well as between ${\widehat{\mathsf{\Phi}}}_{KWW}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{JWS}\left(s\right)$ is shown in Figure 3. It is seen that for large s ${\widehat{\mathsf{\Psi}}}_{JWS}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{JWS}\left(s\right)$ match ${\widehat{\mathsf{\Psi}}}_{KWW}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{KWW}\left(s\right)$ faster than for the CC and HN models. Nevertheless the matchings for small s remain disappointing although at the first glance they seem to be more acceptable than those resulting from the CC and HN models. This, however, may be treated as an artifact coming from the choice of parameters.

- The leading order of large s, i.e., short t, asymptotics of all relaxation patterns being considered matches the KWW function.
- In Figure 1, Figure 2 and Figure 3 the curves labelled by I and II show the behavior of various $\widehat{\mathsf{\Psi}}\left(s\right)$ and $\widehat{\mathsf{\Phi}}\left(s\right)$. Unidexed labels I and II characterize plots obtained for the KWW model if $\alpha =1/3$. The labels I and II indexed with subscripts a, b, and c distinguish non-Debye models: a is for the CC, b is for the HN, and c is for the JWS.

## 5. The Mittag–Leffler Family: Comparison of Useful Properties

#### 5.1. An Interlude: A Few Mathematical Tools

#### 5.1.1. The Efross Theorem as an Integral Decomposition

#### 5.1.2. Integral Decompositions as Subordinations

#### 5.1.3. Subordinations as Signposts Leading to Evolution Equations

#### 5.2. Examples

- (i)
- For the CC model, for which ${\widehat{\mathsf{\Psi}}}_{CC}\left(s\right)$ is given by Equation (14) multiplied by B, we have$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {n}_{CC}\left(t\right)& ={\int}_{0}^{\infty}{\mathrm{e}}^{-Bu}{\mathcal{L}}^{-1}[B{\tau}^{\alpha}{s}^{\alpha -1}{\mathrm{e}}^{-uB{\tau}^{\alpha}{s}^{\alpha}};t]\mathrm{d}u\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\int}_{0}^{\infty}{\mathrm{e}}^{-{\tau}^{-\alpha}\xi}{\mathcal{L}}^{-1}[{s}^{\alpha -1}{\mathrm{e}}^{-\xi {s}^{\alpha}};t]\mathrm{d}\xi ,\hfill \end{array}\end{array}$$$${n}_{CC}\left(t\right)={E}_{\alpha}[-{(t/\tau )}^{\alpha}],$$$${n}_{A;CC}\left(t\right)\sim 1-\frac{{(t/\tau )}^{\alpha}}{\Gamma (1+\alpha )},\phantom{\rule{1.em}{0ex}}t\ll \tau ,\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{n}_{A;CC}\left(t\right)\sim \frac{{(t/\tau )}^{-\alpha}}{\Gamma (1-\alpha )},\phantom{\rule{1.em}{0ex}}t\gg \tau .$$The short time asymptotics given by the first equation in Equation (36) constitutes also the first two terms of the stretched exponential $exp[-{t}^{\alpha}/\Gamma (1+\alpha )]$. Such approximation was proposed in Ref. [49] and it offers good results for low values of $\alpha $ at sufficiently short times.The evolution equations derived from Equation (29) read$${n}_{CC}\left(t\right)=1-{\tau}^{-1}\left({I}^{\alpha}{n}_{CC}\right)\left(t\right)\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\left({}^{c}{D}^{\alpha}{n}_{CC}\right)\left(t\right)=-{\tau}^{\alpha}{n}_{CC}\left(t\right),$$
- (ii)
- Our next example is the HN relaxation function. We begin with the subordination approach which involves the Debye relaxation and ${\mathcal{L}}^{-1}\{\widehat{\mathsf{\Psi}}\left(s\right)exp[-\xi \widehat{\mathsf{\Psi}}\left(s\right)]/s;t\}$. Substituting Equation (16) multiplied by B into Equation (23) we get$${n}_{HN}\left(t\right)=B{\int}_{0}^{\infty}{\mathcal{L}}^{-1}\{B{s}^{-1}[{\tau}^{\alpha \beta}{({\tau}^{-\alpha}+{s}^{\alpha})}^{\beta}-1]{\mathrm{e}}^{-\xi B{\tau}^{\alpha \beta}{({\tau}^{-\alpha}+{s}^{\alpha})}^{\beta}};t\}\mathrm{d}\xi ,$$$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {n}_{HN}\left(t\right)& =B{\int}_{0}^{\infty}\left\{{\int}_{0}^{\infty}{\mathcal{L}}^{-1}[({\tau}^{\alpha \beta}{s}^{\beta}-1){\mathrm{e}}^{-\xi B{\tau}^{\alpha \beta}{s}^{\beta}};u]{\mathrm{e}}^{-u{\tau}^{-\alpha}}{\mathcal{L}}^{-1}[{s}^{-1}{\mathrm{e}}^{-u{s}^{\alpha}};t]\mathrm{d}u\right\}\mathrm{d}\xi \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =B{\int}_{0}^{\infty}{\mathcal{L}}^{-1}\left[({\tau}^{\alpha \beta}{s}^{\beta}-1){\int}_{0}^{\infty}{\mathrm{e}}^{-\xi B{\tau}^{\alpha \beta}{s}^{\beta}}\mathrm{d}\xi ;u\right]{\mathcal{L}}^{-1}[{s}^{-1}{\mathrm{e}}^{-u({\tau}^{-\alpha}+{s}^{\alpha})};t]\mathrm{d}u\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\int}_{0}^{\infty}{\mathrm{e}}^{-u{\tau}^{-\alpha}}{\mathcal{L}}^{-1}[{s}^{-\beta}({s}^{\beta}-{\tau}^{-\alpha \beta});u]{\mathcal{L}}^{-1}[{s}^{-1}{\mathrm{e}}^{-u{s}^{\alpha}};t]\mathrm{d}u.\hfill \end{array}\end{array}$$Because of ${\mathcal{L}}^{-1}[{s}^{-\beta}({s}^{\beta}-{\tau}^{-\alpha \beta});u]=\delta \left(u\right)-{\tau}^{-\alpha \beta}{u}^{\beta -1}/\Gamma \left(\beta \right)$ we rewrite Equation (39) as$$\begin{array}{cc}\hfill {n}_{HN}\left(t\right)& =1-{\tau}^{-\alpha \beta}{\int}_{0}^{\infty}{\mathrm{e}}^{-{\tau}^{-\alpha}u}\frac{{u}^{\beta -1}}{\Gamma \left(\beta \right)}\phantom{\rule{0.166667em}{0ex}}{\mathcal{L}}^{-1}[{s}^{-1}{\mathrm{e}}^{-u{s}^{\alpha}};t]\mathrm{d}u\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =1-{(t/\tau )}^{\alpha \beta}{\int}_{0}^{\infty}{\mathrm{e}}^{-{\tau}^{-\alpha}u}\phantom{\rule{0.166667em}{0ex}}\frac{t}{\alpha u}{g}_{\alpha ,1+\alpha \beta}^{\beta}(u,t)\mathrm{d}u.\hfill \end{array}$$From Equation (33) it comes out that ${\mathcal{L}}^{-1}[{s}^{-1}{\mathrm{e}}^{-u{s}^{\alpha}};t]=\Gamma \left(\beta \right){t}^{1+\alpha \beta}/\left(\alpha {u}^{\beta}\right){g}_{\alpha ,1+\alpha \beta}^{\beta}(u,t)$. Then, with the help of Equation (A4), we get$${n}_{HN}\left(t\right)=1-{(t/\tau )}^{\alpha \beta}{E}_{\alpha ,1+\alpha \beta}^{\beta}[-{(t/\tau )}^{\alpha}],$$$${\mathrm{e}}^{-{\tau}^{-\alpha}u}\frac{{u}^{\beta -1}}{\Gamma \left(\beta \right)}={\tau}^{\alpha \beta}{\mathcal{L}}^{-1}[{(1+s{\tau}^{\alpha})}^{-\beta};t],$$$${n}_{A;HN}\left(t\right)\sim 1-\frac{{(t/\tau )}^{\alpha \beta}}{\Gamma (1+\alpha \beta )},\phantom{\rule{1.em}{0ex}}t\ll \tau ,\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{n}_{A;HN}\left(t\right)\sim \frac{\beta \phantom{\rule{0.166667em}{0ex}}{(t/\tau )}^{-\alpha}}{\Gamma (1-\alpha )},\phantom{\rule{1.em}{0ex}}t\gg \tau .$$The evolution equation is equal to ([18] Equations (3.40) and (3.50))$${}^{\mathrm{C}}{({D}^{\alpha}+{\tau}^{-\alpha})}^{\beta}{n}_{HN}\left(t\right)=-{\tau}^{\alpha \beta},$$
- (iii)
- Equation (23) for the JWS model gives$${n}_{JWS}\left(t\right)=B{\int}_{0}^{\infty}{\mathcal{L}}^{-1}\left[\frac{{s}^{\alpha \beta -1}}{{({s}^{\alpha}+{\tau}^{-\alpha})}^{\beta}-{s}^{\alpha \beta}}{\mathrm{e}}^{-\xi B\frac{{({s}^{\alpha}+{\tau}^{-\alpha})}^{\beta}}{{({s}^{\alpha}+{\tau}^{-\alpha})}^{\beta}-{s}^{\alpha \beta}}};t\right]\mathrm{d}\xi ,$$$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {n}_{JWS}\left(t\right)& =B{\int}_{0}^{\infty}\left\{{\int}_{0}^{\infty}{\mathcal{L}}^{-1}\left[\frac{1}{{s}^{\beta}-{(s-{\tau}^{-\alpha})}^{\beta}}{\mathrm{e}}^{-\xi B\frac{{s}^{\beta}}{{s}^{\beta}-{(s-{\tau}^{-\alpha})}^{\beta}}};u\right]\right.\hfill \\ & \left.\times {\mathcal{L}}^{-1}[{s}^{\alpha \beta -1}{\mathrm{e}}^{-u({\tau}^{-\alpha}+{s}^{\alpha})};t]\mathrm{d}u\right\}\mathrm{d}\xi .\hfill \end{array}\end{array}$$

**ii**)) we can simplify the first inverse Laplace transform. It enables us to write down the above equation in the form

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

KWW | Kohlrausch–Williams–Watts relaxation |

CC | Cole–Cole relaxation |

CD | Cole–Davidson relaxation |

HN | Havriliak–Negami relaxation |

JWS | Jurlewicz–Weron–Stanislavsky relaxation |

CMF | Completely monotone function |

SF | Stieltjes function |

CBF | Completely Bernstein function |

## Appendix A. The Stretched Exponential and Mittag–Leffler Functions

## Appendix B. The Completely Monotone and Completely Bernstein Functions

## Appendix C. The Fox H, Meijer G, and Generalized Hypergeometric Functions

## Appendix D. Relation between the Infinitely Divisible Distribution and the Bernstein-Class Functions

## Appendix E. Fractional Integrals and Derivatives

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**Figure 1.**Plot presents the comparison between characteristic functions ${\widehat{\mathsf{\Psi}}}_{KWW}\left(s\right)$ (red solid curve no. I) and ${\widehat{\mathsf{\Psi}}}_{CC}\left(s\right)$ (brown dot-dashed curve no. Ia), as well as between partner functions ${\widehat{\mathsf{\Phi}}}_{KWW}\left(s\right)$ (blue solid curve no. II) and ${\widehat{\mathsf{\Phi}}}_{CC}\left(s\right)$ (green dashed curve no. IIa) for $\alpha =1/3$ and $\tau =1$. The characteristic exponents ${\widehat{\mathsf{\Psi}}}_{KWW}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{KWW}\left(s\right)$ have been calculated using Equation (8) whereas ${\widehat{\mathsf{\Psi}}}_{CC}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{CC}\left(s\right)$ are given by Equation (14). The latter ones are, respectively, multiplied and divided by ${[\Gamma (4/3)]}^{-1}$.

**Figure 2.**Plot presents the comparison between characteristic exponents ${\widehat{\mathsf{\Psi}}}_{KWW}\left(s\right)$ (red solid curve no. I) and ${\widehat{\mathsf{\Psi}}}_{HN}\left(s\right)$ (brown dot-dashed curve no. Ib), as well as between partner functions ${\widehat{\mathsf{\Phi}}}_{KWW}\left(s\right)$ (blue solid curve no. II) and ${\widehat{\mathsf{\Phi}}}_{HN}\left(s\right)$ (green dashed curve no. IIb). ${\widehat{\mathsf{\Psi}}}_{KWW}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{KWW}\left(s\right)$ are calculated with the help of Equation (8) where we use ${\alpha}_{KWW}=1/3$, $\tau =1$. The characteristic exponents ${\widehat{\mathsf{\Psi}}}_{HN}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{HN}\left(s\right)$ are given by Equation (16) where ${\alpha}_{HN}=5/6$ and ${\beta}_{HN}=2/5$. The comparison between the characteristic functions is made with the factor ${[\Gamma (1+5/6)]}^{-1}$ which is multiplied by ${\widehat{\mathsf{\Psi}}}_{HN}\left(s\right)$ and divided by ${\widehat{\mathsf{\Phi}}}_{HN}\left(s\right)$.

**Figure 3.**Plot presents the comparison between ${\widehat{\mathsf{\Psi}}}_{KWW}\left(s\right)$ (red solid curve no. I) and ${\widehat{\mathsf{\Psi}}}_{JWS}\left(s\right)$ (brown dot-dashed curve no. Ic), as well as between ${\widehat{\mathsf{\Phi}}}_{KWW}\left(s\right)$ (blue solid curve no. II) and ${\widehat{\mathsf{\Phi}}}_{JWS}\left(s\right)$ (green dashed curve no. IIc). ${\widehat{\mathsf{\Psi}}}_{KWW}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{KWW}\left(s\right)$ are calculated with the help of Equation (8) where we use ${\alpha}_{KWW}=1/3$, $\tau =1$. The characteristic exponents ${\widehat{\mathsf{\Psi}}}_{JWS}\left(s\right)$ and ${\widehat{\mathsf{\Phi}}}_{JWS}\left(s\right)$ are given by Equations (16) where ${\alpha}_{JWS}=2/5$, ${\beta}_{JWS}=5/6$, and are, respectively, multiplied and divided by ${[\Gamma (1+2/5)]}^{-1}$.

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**MDPI and ACS Style**

Górska, K.; Horzela, A.; Penson, K.A.
Non-Debye Relaxations: The Ups and Downs of the Stretched Exponential vs. Mittag–Leffler’s Matchings. *Fractal Fract.* **2021**, *5*, 265.
https://doi.org/10.3390/fractalfract5040265

**AMA Style**

Górska K, Horzela A, Penson KA.
Non-Debye Relaxations: The Ups and Downs of the Stretched Exponential vs. Mittag–Leffler’s Matchings. *Fractal and Fractional*. 2021; 5(4):265.
https://doi.org/10.3390/fractalfract5040265

**Chicago/Turabian Style**

Górska, Katarzyna, Andrzej Horzela, and Karol A. Penson.
2021. "Non-Debye Relaxations: The Ups and Downs of the Stretched Exponential vs. Mittag–Leffler’s Matchings" *Fractal and Fractional* 5, no. 4: 265.
https://doi.org/10.3390/fractalfract5040265