# Variable-Order Fractional Scale Calculus

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. On Scale-Invariant Systems: Hadamard Derivatives

**Definition 1.**

**Definition 2.**

**Theorem 1**

**.**The powers ${\tau}^{v},\phantom{\rule{0.166667em}{0ex}}\tau \in {\mathbb{R}}^{+},v\in \mathbb{C}$ are the eigenfunctions of the DI systems: if $x\left(\tau \right)={\tau}^{v}$, then

**Remark 1.**

**Definition 3.**

**Definition 4.**

**Corollary 1.**

**Theorem 2.**

**Proof.**

## 3. Variable-Order Scale Derivatives

#### 3.1. GL-Like Derivatives

**Definition 5.**

**Theorem 3.**

**Proof.**

**Definition 6.**

#### 3.2. VO Hadamard Derivatives

**Theorem 4.**

**Proof.**

**Definition 7.**

**Definition 8.**

#### 3.3. Derivative Properties

- This integral is not an inverse MT;
- This integral shows that ${\mathfrak{D}}_{s\pm}^{\alpha \left(\tau \right)}x\left(\tau \right)$ can be considered a synthesis of elemental powers ${v}^{\alpha \left(\tau \right)}X\left(v\right){\tau}^{v}\mathrm{d}v$, which provides it sense and meaning;
- The expression ${v}^{\alpha \left(\tau \right)}X\left(v\right)$ is not an MT. In fact, its transform is given by:$${\overline{X}}_{\alpha}\left(u\right)=\mathcal{M}\left[{\mathfrak{D}}_{s\pm}^{\alpha \left(\tau \right)}x\left(\tau \right)\right]\left(u\right)=\frac{1}{2\pi i}\underset{\sigma -i\infty}{\overset{\sigma +i\infty}{\int}}\mathcal{M}\left[{v}^{\alpha \left(\tau \right)}{\tau}^{v}\right]X\left(v\right)\mathrm{d}v,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\tau \in {\mathbb{R}}^{+}.$$

- In general,$$\begin{array}{c}\hfill {\mathfrak{D}}_{s\pm}^{{\alpha}_{1}\left(\tau \right)+{\alpha}_{2}\left(\tau \right)}x\left(\tau \right)\ne {\mathfrak{D}}_{s\pm}^{{\alpha}_{2}\left(\tau \right)}{\mathfrak{D}}_{s\pm}^{{\alpha}_{1}\left(\tau \right)}x\left(\tau \right);\end{array}$$
- There is no recursivity$$\begin{array}{c}\hfill {\mathfrak{D}}_{s\pm}^{n\alpha \left(\tau \right)}x\left(\tau \right)\ne {\mathfrak{D}}_{s\pm}^{\alpha \left(\tau \right)}{\mathfrak{D}}_{s\pm}^{(n-1)\alpha \left(\tau \right)}x\left(\tau \right);\end{array}$$
- If ${\alpha}_{1}\left(\tau \right)$ is a real constant ${\alpha}_{1}\left(\tau \right)={\alpha}_{0}$, then ${\overline{X}}_{{\alpha}_{0}}\left(u\right)={u}^{{\alpha}_{0}}X\left(u\right)$ and$$\begin{array}{c}\hfill {\mathfrak{D}}_{s\pm}^{{\alpha}_{0}+{\alpha}_{2}\left(\tau \right)}x\left(\tau \right)={\mathfrak{D}}_{s\pm}^{{\alpha}_{2}\left(\tau \right)}{\mathfrak{D}}_{s\pm}^{{\alpha}_{0}}x\left(\tau \right);\end{array}$$
- However,$$\begin{array}{c}\hfill {\mathfrak{D}}_{s\pm}^{{\alpha}_{0}}{\mathfrak{D}}_{s\pm}^{{\alpha}_{2}\left(\tau \right)}x\left(\tau \right)\ne {\mathfrak{D}}_{s\pm}^{{\alpha}_{2}\left(\tau \right)}{\mathfrak{D}}_{s\pm}^{{\alpha}_{0}}x\left(\tau \right),\end{array}$$
- As it becomes clear, the anti-derivative of ${\mathfrak{D}}_{s\pm}^{\alpha \left(\tau \right)}$ is not ${\mathfrak{D}}_{s\pm}^{-\alpha \left(\tau \right)}.$

**Examples.**Let

## 4. VO Scale-Invariant Systems

**Definition 9.**

**Definition 10.**

**Remark 2.**

**Examples.**Let

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Nottale, L. The theory of scale relativity. Int. J. Mod. Phys.
**1992**, 7, 4899–4936. [Google Scholar] [CrossRef] - Cohen, L. The scale representation. IEEE Trans. Signal Process.
**1993**, 41, 3275–3292. [Google Scholar] [CrossRef] - Proekt, A.; Banavar, J.R.; Maritan, A.; Pfaff, D.W. Scale invariance in the dynamics of spontaneous behavior. Proc. Natl. Acad. Sci. USA
**2012**, 109, 10564–10569. [Google Scholar] [CrossRef] [PubMed] - Khaluf, Y.; Ferrante, E.; Simoens, P.; Huepe, C. Scale invariance in natural and artificial collective systems: A review. J. R. Soc. Interface
**2017**, 14, 20170662. [Google Scholar] [CrossRef] - Lamperti, J. Semi-stable stochastic processes. Trans. Am. Math. Soc.
**1962**, 104, 62–78. [Google Scholar] [CrossRef] - Borgnat, P.; Amblard, P.O.; Flandrin, P. Scale invariances and Lamperti transformations for stochastic processes. J. Phys. Math. Gen.
**2005**, 38, 2081. [Google Scholar] [CrossRef] - Belbahri, K. Scale invariant operators and combinatorial expansions. Adv. Appl. Math.
**2010**, 45, 548–563. [Google Scholar] [CrossRef] - Nottale, L. Non-differentiable space-time and scale relativity. In Proceedings of the International Colloquium Géométrie au XXe Siècle, Paris, France, 24–29 September 2001. [Google Scholar]
- Nottale, L. The Theory of Scale Relativity: Non-Differentiable Geometry and Fractal Space-Time. In Proceedings of the AIP Conference Proceedings; American Institute of Physics: College Park, MD, USA, 2004; Volume 718, pp. 68–95. [Google Scholar]
- Cresson, J. Scale relativity theory for one-dimensional non-differentiable manifolds. Chaos Solitons Fractals
**2002**, 14, 553–562. [Google Scholar] [CrossRef] - Cresson, J. Scale calculus and the Schrödinger equation. J. Math. Phys.
**2003**, 44, 4907–4938. [Google Scholar] [CrossRef] - Scher, H.; Shlesinger, M.F.; Bendler, J.T. Time-scale invariance in transport and relaxation. Phys. Today
**1991**, 44, 26–34. [Google Scholar] [CrossRef] - Yulmetyev, R.M.; Mokshin, A.V.; Hänggi, P.; Shurygin, V.Y. Time-scale invariance of relaxation processes of density fluctuation in slow neutron scattering in liquid cesium. Phys. Rev.
**2001**, 64, 057101. [Google Scholar] [CrossRef] - Stanley, H.E.; Amaral, L.A.N.; Gopikrishnan, P.; Plerou, V.; Salinger, M.A. Scale invariance and universality in economic phenomena. J. Phys. Condens. Matter
**2002**, 14, 2121. [Google Scholar] [CrossRef] - James, A.F.; Peter, G.F. Discrete scale invariance in stock markets before crashes. Int. J. Mod. Phys. B
**1996**, 10, 3737–3745. [Google Scholar] - Buhusi, C.V.; Oprisan, S.A. Time-scale invariance as an emergent property in a perceptron with realistic, noisy neurons. Behav. Process.
**2013**, 95, 60–70. [Google Scholar] [CrossRef] - Grossmann, A.; Morlet, J. Decomposition of Hardy functions into square integrable wavelets of constant shape. Siam J. Math. Anal.
**1984**, 15, 723–736. [Google Scholar] [CrossRef] - Meyer, Y. Orthonormal wavelets. In Proceedings of the Wavelets: Time-Frequency Methods and Phase Space Proceedings of the International Conference, Marseille, France, 14–18 December 1987; pp. 21–37.
- Mallat, S.G. Multiresolution Representations and Wavelets; University of Pennsylvania: Philadelphia, PA, USA, 1988. [Google Scholar]
- Mallat, S. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell.
**1989**, 11, 674–693. [Google Scholar] [CrossRef] - Daubechies, I. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory
**1990**, 36, 961–1005. [Google Scholar] [CrossRef] - Braccini, C.; Gambardella, G. Form-invariant linear filtering: Theory and applications. IEEE Trans. Acoust. Speech Signal Process.
**1986**, 34, 1612–1628. [Google Scholar] [CrossRef] - Yazici, B.; Kashyap, R.L. A class of second-order stationary self-similar processes for 1/f phenomena. IEEE Trans. Signal Process.
**1997**, 45, 396–410. [Google Scholar] [CrossRef] - Ortigueira, M.D. On the Fractional Linear Scale Invariant Systems. IEEE Trans. Signal Process.
**2010**, 58, 6406–6410. [Google Scholar] [CrossRef] - Tarasov, V.E. Fractional dynamics with non-local scaling. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 102, 105947. [Google Scholar] [CrossRef] - Ortigueira, M.D.; Bohannan, G.W. Fractional Scale Calculus: Hadamard vs. Liouville. Fractal Fract.
**2023**, 7, 296. [Google Scholar] [CrossRef] - Tarasov, V.E. Scale-Invariant General Fractional Calculus: Mellin Convolution Operators. Fractal Fract.
**2023**, 7, 481. [Google Scholar] [CrossRef] - Hadamard, J. Essai sur L’étude des Fonctions, Données par leur Développement de Taylor; Gallica: Paris, France, 1892; pp. 101–186. [Google Scholar]
- Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives; Gordon and Breach: Yverdon, Switzerland, 1993. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Jarad, F.; Abdeljawad, T.; Baleanu, D. Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ.
**2012**, 2012, 1–8. [Google Scholar] [CrossRef] - Benkerrouche, A.; Souid, M.S.; Stamov, G.; Stamova, I. Multiterm Impulsive Caputo–Hadamard Type Differential Equations of Fractional Variable Order. Axioms
**2022**, 11, 634. [Google Scholar] [CrossRef] - Ma, L.; Li, C. On Hadamard fractional calculus. Fractals
**2017**, 25, 1750033. [Google Scholar] [CrossRef] - Samko, S.G.; Ross, B. Integration and differentiation to a variable fractional order. Integral Transform. Spec. Funct.
**1993**, 4, 277–300. [Google Scholar] [CrossRef] - Ross, B.; Samko, S. Fractional integration operator of variable order in the holder spaces H
^{λ(x)}. Int. J. Math. Math. Sci.**1995**, 18, 777–788. [Google Scholar] [CrossRef] - Samko, S. Fractional integration and differentiation of variable order: An overview. Nonlinear Dyn.
**2013**, 71, 653–662. [Google Scholar] [CrossRef] - Lorenzo, C.F.; Hartley, T.T. Variable Order and Distributed Order Fractional Operators. Nonlinear Dyn.
**2002**, 29, 57–98. [Google Scholar] [CrossRef] - Coimbra, C. Mechanics with variable-order differential operators. Ann. Der Phys.
**2003**, 515, 692–703. [Google Scholar] [CrossRef] - Valério, D.; Sá da Costa, J. Variable-order fractional derivatives and their numerical approximations. Signal Process.
**2011**, 91, 470–483. [Google Scholar] [CrossRef] - Sierociuk, D.; Malesza, W.; Macias, M. Derivation, interpretation, and analog modelling of fractional variable order derivative definition. Appl. Math. Model.
**2015**, 39, 3876–3888. [Google Scholar] [CrossRef] - Sierociuk, D.; Malesza, W.; Macias, M. On the Recursive Fractional Variable-Order Derivative: Equivalent Switching Strategy, Duality, and Analog Modeling. Circuits Syst. Signal Process.
**2015**, 34, 1077–1113. [Google Scholar] [CrossRef] - Sierociuk, D.; Malesza, W. On the differences of variable type and variable fractional order. In Proceedings of the 2016 European Control Conference (ECC), Aalborg, Denmark, 29 June–1 July 2016. [Google Scholar]
- Sierociuk, D.; Malesza, W. Fractional variable order discrete-time systems, their solutions and properties. Int. J. Syst. Sci.
**2017**, 48, 3098–3105. [Google Scholar] [CrossRef] - Ortigueira, M.D.; Valério, D.; Machado, J.A.T. Variable order fractional systems. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 71, 231–243. [Google Scholar] [CrossRef] - Ortigueira, M.D.; Machado, J.A.T. Fractional Derivatives: The Perspective of System Theory. Mathematics
**2019**, 7, 150. [Google Scholar] [CrossRef] - Valério, D.; Ortigueira, M.D.; Lopes, A.M. How Many Fractional Derivatives Are There? Mathematics
**2022**, 10, 737. [Google Scholar] [CrossRef] - Bertrand, J.; Bertrand, P.; Ovarlez, J.P. The Mellin Transform. In The Transformsand Applications Handbook, 3rd ed.; Poularikas, A.D., Grigoryan, A.M., Eds.; CRC Press: Boca Raton, FL, USA, 2018; Chapter 12. [Google Scholar]
- Butzer, P.L.; Jansche, S. A direct approach to the Mellin transform. J. Fourier Anal. Appl.
**1997**, 3, 325–376. [Google Scholar] [CrossRef] - Poularikas, A.D. The Transforms and Applications Handbook; CRC Press LLC: Boca Raton, FL, USA, 2000. [Google Scholar]
- Luchko, Y.; Kiryakova, V. The Mellin integral transform in fractional calculus. Fract. Calc. Appl. Anal.
**2013**, 16, 405–430. [Google Scholar] [CrossRef] - Ortigueira, M.D.; Machado, J.T. Revisiting the 1D and 2D Laplace transforms. Mathematics
**2020**, 8, 1330. [Google Scholar] [CrossRef] - Almeida, R.; Torres, D.F. Computing Hadamard type operators of variable fractional order. Appl. Math. Comput.
**2015**, 257, 74–88. [Google Scholar] [CrossRef] - Almeida, R. Caputo–Hadamard fractional derivatives of variable order. Numer. Funct. Anal. Optim.
**2017**, 38, 1–19. [Google Scholar] [CrossRef] - Zheng, X. Logarithmic transformation between (variable-order) Caputo and Caputo–Hadamard fractional problems and applications. Appl. Math. Lett.
**2021**, 121, 107366. [Google Scholar] [CrossRef] - Garrappa, R.; Giusti, A.; Mainardi, F. Variable-order fractional calculus: A change of perspective. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 102, 105904. [Google Scholar] [CrossRef] - Ortigueira, M.D.; Machado, J.T. What is a fractional derivative? J. Comput. Phys.
**2015**, 293, 4–13. [Google Scholar] [CrossRef] - Ortigueira, M.D.; Bengochea, G.; Machado, J.A.T. Substantial, tempered, and shifted fractional derivatives: Three faces of a tetrahedron. Math. Methods Appl. Sci.
**2021**, 44, 9191–9209. [Google Scholar] [CrossRef] - Ortigueira, M.D.; Magin, R.L. On the Equivalence between Integer-and Fractional Order-Models of Continuous-Time and Discrete-Time ARMA Systems. Fractal Fract.
**2022**, 6, 242. [Google Scholar] [CrossRef] - Bengochea, G.; Ortigueira, M.; Verde-Star, L. Operational calculus for the solution of fractional differential equations with noncommensurate orders. Math. Methods Appl. Sci.
**2021**, 44, 8088–8096. [Google Scholar] [CrossRef] - Ortigueira, M.D.; Valério, D. Fractional Signals and Systems; De Gruyter: Berlin, Germany; Berlin, MA, USA, 2020. [Google Scholar]
- Bengochea, G.; Ortigueira, M.; Verde-Star, L. The causal α-exponential and the solution of fractional linear time-invariant systems. Math. Methods Appl. Sci. 2023; Submitted. [Google Scholar]

**Figure 1.**Responses of (62), for both the stretch (blue) and the shrink (red) cases.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Valério, D.; Ortigueira, M.D.
Variable-Order Fractional Scale Calculus. *Mathematics* **2023**, *11*, 4549.
https://doi.org/10.3390/math11214549

**AMA Style**

Valério D, Ortigueira MD.
Variable-Order Fractional Scale Calculus. *Mathematics*. 2023; 11(21):4549.
https://doi.org/10.3390/math11214549

**Chicago/Turabian Style**

Valério, Duarte, and Manuel D. Ortigueira.
2023. "Variable-Order Fractional Scale Calculus" *Mathematics* 11, no. 21: 4549.
https://doi.org/10.3390/math11214549