# Fractional Velocity as a Tool for the Study of Non-Linear Problems

## Abstract

**:**

## 1. Introduction

## 2. Fractional Variations and Fractional Velocities

**Definition**

**1.**

**Definition**

**2**

**.**Define the fractional velocity of fractional order β as the limit

**Condition**

**1**

**.**For given x and $0<\beta \le 1$

**Condition**

**2**

**.**For given x, $0<\beta \le 1$ and $\u03f5>0$

**Theorem**

**1**

**.**For each $\beta >0$ if ${\upsilon}_{+}^{\beta}f\left(x\right)$ exists (finitely), then f is right-Hölder continuous of order β at x and C1 holds, and the analogous result holds for ${\upsilon}_{-}^{\beta}f\left(x\right)$ and left-Hölder continuity.

**Proposition**

**1**

**.**The existence of ${\upsilon}_{\pm}^{\beta}f\left(x\right)\ne 0$ for $\beta \le 1$ implies that

**Remark**

**1.**

**Proposition**

**2.**

**Proof.**

**Remark**

**2.**

## 3. Characterization of Singular Functions

#### 3.1. Scale Embedding of Fractional Velocities

**Definition**

**3.**

**set of change**${\chi}_{\pm}^{\beta}\left(f\right):=\left\{x:{\upsilon}_{\pm}^{\beta}f\left(x\right)\ne 0\right\}$.

**scale-dependent operators**for a wide variety if physical signals. An extreme case of such signals are the singular functions

**Proposition**

**3.**

**Theorem**

**2**

**.**Suppose that $f\in BVC[x,x+\u03f5]$ and ${f}^{\prime}$ does not vanish a.e. in $[x,x+\u03f5]$. Suppose that $\varphi \in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}1}$ is a contraction map. Let ${f}_{n}\left(x\right):={\underbrace{\varphi \circ \dots \varphi}}_{n}\circ f\left(x\right)$ be the n-fold composition and $F\left(x\right):=\underset{n\to \infty}{\mathrm{lim}}{f}_{n}\left(x\right)$ exists finitely. Then the following commuting diagram holds:

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**scale–regularizing**sequence.

**Proof.**

#### 3.2. Applications

#### 3.2.1. De Rham Function

**Proposition**

**4.**

#### 3.2.2. Bernoulli-Mandelbrot Binomial Measure

**Remark**

**3.**

#### 3.2.3. Neidinger Function

#### 3.2.4. Langevin Evolution

#### 3.2.5. Brownian Motion

## 4. Characterization of Kolwankar-Gangal Local Fractional Derivatives

#### 4.1. Fractional Integrals and Derivatives

**Definition**

**4.**

**Example**

**1.**

#### 4.2. The Local(ized) Fractional Derivative

**Definition**

**5.**

**Remark**

**4.**

**Proposition**

**5**

**.**Let $f\left(x\right)$ be β-differentiable about x. Then ${\mathcal{D}}_{KG,\pm}^{\beta}f\left(x\right)$ exists and

**Proof.**

**Proposition**

**6.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**3**

**.**For $0<\beta <1$ if ${\mathcal{D}}_{KG\pm}^{\beta}f\left(x\right)$ is continuous about x then ${\mathcal{D}}_{KG\pm}^{\beta}f\left(x\right)=0$.

**Proof.**

**Corollary**

**3**

**.**Let ${\chi}_{\beta}:=\{x:{\mathcal{D}}_{KG\pm}^{\beta}f\left(x\right)\ne 0\}$. Then for $0<\beta <1$ ${\chi}_{\beta}$ is totally disconnected.

**Remark**

**5.**

#### 4.3. Equivalent Forms of LFD

**Proposition**

**7.**

**Proposition**

**8.**

## 5. Discussion

## 6. Conclusions

**local non-linear behavior**of functions as demonstrated by the presented examples. In applied problems, local fractional derivatives can be also used to derive fractional Taylor expansions [24,42,43].

## Acknowledgments

## Conflicts of Interest

## Appendix A. General Definitions and Notations

**Definition**

**A1.**

**Definition**

**A2**

**.**The notation ${\scriptstyle \mathcal{O}}\left({x}^{\alpha}\right)$ is interpreted as the convention that

**Definition**

**A3.**

**F-analytic**functions.

**Definition**

**A4.**

**Remark**

**A1.**

**Definition**

**A5.**

**Definition**

**A6.**

## Appendix B. Essential Properties of Fractional Velocity

- Product rule$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}\left[f\phantom{\rule{0.166667em}{0ex}}g\right]\left(x\right)& ={\upsilon}_{+}^{\beta}f\left(x\right)g\left(x\right)+{\upsilon}_{+}^{\beta}g\left(x\right)f\left(x\right)+{[f,g]}_{\beta}^{+}\left(x\right)\hfill \\ \hfill {\upsilon}_{-}^{\beta}\left[f\phantom{\rule{0.166667em}{0ex}}g\right]\left(x\right)& ={\upsilon}_{-}^{\beta}f\left(x\right)g\left(x\right)+{\upsilon}_{-}^{\beta}g\left(x\right)f\left(x\right)-{[f,g]}_{\beta}^{-}\left(x\right)\hfill \end{array}$$
- Quotient rule$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}[f/g]\left(x\right)& =\frac{{\upsilon}_{+}^{\beta}f\left(x\right)g\left(x\right)-{\upsilon}_{+}^{\beta}g\left(x\right)f\left(x\right)-{[f,g]}_{\beta}^{+}}{{g}^{2}\left(x\right)}\hfill \\ \hfill {\upsilon}_{-}^{\beta}[f/g]\left(x\right)& =\frac{{\upsilon}_{-}^{\beta}f\left(x\right)g\left(x\right)-{\upsilon}_{-}^{\beta}g\left(x\right)f\left(x\right)+{[f,g]}_{\beta}^{-}}{{g}^{2}\left(x\right)}\hfill \end{array}$$$${[f,g]}_{\beta}^{\pm}\left(x\right):=\underset{\u03f5\to 0}{\mathrm{lim}}{\upsilon}_{\u03f5\pm}^{\beta /2}\left[fg\right]\left(x\right)$$

- $f\in {\mathbb{H}}^{\phantom{\rule{0.166667em}{0ex}}\beta}$ and $g\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}1}$$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}f\circ g\left(x\right)& ={\upsilon}_{+}^{\beta}f\left(g\right){\left({g}^{\prime}\left(x\right)\right)}^{\beta}\hfill \\ \hfill {\upsilon}_{-}^{\beta}f\circ g\left(x\right)& ={\upsilon}_{-}^{\beta}f\left(g\right){\left({g}^{\prime}\left(x\right)\right)}^{\beta}\hfill \end{array}$$
- $f\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}1}$ and $g\in {\mathbb{H}}^{\phantom{\rule{0.166667em}{0ex}}\beta}$$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}f\circ g\left(x\right)& ={f}^{\prime}\left(g\right)\phantom{\rule{0.166667em}{0ex}}{\upsilon}_{+}^{\beta}g\left(x\right)\hfill \\ \hfill {\upsilon}_{-}^{\beta}f\circ g\left(x\right)& ={f}^{\prime}\left(g\right)\phantom{\rule{0.166667em}{0ex}}{\upsilon}_{-}^{\beta}g\left(x\right)\hfill \end{array}$$

## References

- Mandelbrot, B. Fractal Geometry of Nature; Henry Holt & Co.: New York, NY, USA, 1982. [Google Scholar]
- Mandelbrot, B. Les Objets Fractals: Forme, Hasard et Dimension; Flammarion: Paris, France, 1989. [Google Scholar]
- Metzler, R.; Klafter, J. The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A Math. Gen.
**2004**, 37, R161–R208. [Google Scholar] [CrossRef] - Wheatcraft, S.W.; Meerschaert, M.M. Fractional conservation of mass. Adv. Water Resour.
**2008**, 31, 1377–1381. [Google Scholar] [CrossRef] - Caputo, M.; Mainardi, F. Linear models of dissipation in anelastic solids. Rivista del Nuovo Cimento
**1971**, 1, 161–198. [Google Scholar] [CrossRef] - Mainardi, F. Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics. In Fractals and Fractional Calculus in Continuum Mechanics; Springer: Wien, Austria; New York, NY, USA, 1997; pp. 291–348. [Google Scholar]
- Gorenflo, R.; Mainardi, F. Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: Mathematical aspects. In Anomalous Transport; Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2008; pp. 93–127. [Google Scholar]
- Oldham, K.B.; Spanier, J.S. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order; Academic Press: New York, NY, USA, 1974. [Google Scholar]
- Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise; Dover Publications: Mineola, NY, USA, 1991. [Google Scholar]
- Losa, G.; Nonnenmacher, T. Self-similarity and fractal irregularity in pathologic tissues. Mod. Pathol.
**1996**, 9, 174–182. [Google Scholar] [PubMed] - Darst, R.; Palagallo, J.; Price, T. Curious Curves; World Scientific Publishing Company: Singapore, 2009. [Google Scholar]
- John Hutchinson. Fractals and self similarity. Indiana Univ. Math. J.
**1981**, 30, 713–747. [Google Scholar] - Mandelbro, B.B. Intermittent Turbulence in Self-Similar Cascades: Divergence of High Moments and Dimension of the Carrier; Springer: New York, NY, USA, 1999; pp. 317–357. [Google Scholar]
- Meneveau, C.; Sreenivasan, K.R. Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett.
**1987**, 59, 1424–1427. [Google Scholar] - Sreenivasan, K.R.; Meneveau, C. The fractal facets of turbulence. J. Fluid Mech.
**1986**, 173, 357–386. [Google Scholar] [CrossRef] - Puente, C.; López, M.; Pinzón, J.; Angulo, J. The gaussian distribution revisited. Adv. Appl. Probab.
**1996**, 28, 500–524. [Google Scholar] [CrossRef] - Nottale, L. Scale relativity and fractal space-time: Theory and applications. Found. Sci.
**2010**, 15, 101–152. [Google Scholar] [CrossRef] - Cresson, J.; Pierret, F. Multiscale functions, scale dynamics, and applications to partial differential equations. J. Math. Phys.
**2016**, 57, 053504. [Google Scholar] [CrossRef] - Cherbit, G. Local dimension, momentum and trajectories. In Fractals, Non-Integral Dimensions and Applications; John Wiley & Sons: Paris, France, 1991; pp. 231–238. [Google Scholar]
- Prodanov, D. Characterization of strongly non-linear and singular functions by scale space analysis. Chaos Solitons Fractals
**2016**, 93, 14–19. [Google Scholar] [CrossRef] - Du Bois-Reymond, P. Versuch einer classification der willkürlichen functionen reeller argumente nach ihren aenderungen in den kleinsten intervallen. J. Reine Angew. Math.
**1875**, 79, 21–37. [Google Scholar] - Faber, G. Über stetige funktionen. Math. Ann.
**1909**, 66, 81–94. [Google Scholar] [CrossRef] - Ben Adda, F.; Cresson, J. About non-differentiable functions. J. Math. Anal. Appl.
**2001**, 263, 721–737. [Google Scholar] [CrossRef] - Prodanov, D. Conditions for continuity of fractional velocity and existence of fractional Taylor expansions. Chaos Solitons Fractals
**2017**, 102, 236–244. [Google Scholar] [CrossRef] - Odibat, Z.M.; Shawagfeh, N.T. Generalized Taylor’s formula. Appl. Math. Comput.
**2007**, 186, 286–293. [Google Scholar] [CrossRef] - Chen, Y.; Yan, Y.; Zhang, K. On the local fractional derivative. J. Math. Anal. Appl.
**2010**, 362, 17–33. [Google Scholar] [CrossRef] - Lomnicki, Z.; Ulam, S. Sur la théorie de la mesure dans les espaces combinatoires et son application au calcul des probabilités i. variables indépendantes. Fundam. Math.
**1934**, 23, 237–278. [Google Scholar] [CrossRef] - Cesàro, E. Fonctions continues sans dérivée. Arch. Math. Phys.
**1906**, 10, 57–63. [Google Scholar] - Salem, R. On some singular monotonic functions which are strictly increasing. Trans. Am. Math. Soc.
**1943**, 53, 427–439. [Google Scholar] - Berg, L.; Krüppel, M. De rham’s singular function and related functions. Zeitschrift für Analysis und Ihre Anwendungen
**2000**, 19, 227–237. [Google Scholar] - Neidinger, R. A fair-bold gambling function is simply singular. Am. Math. Mon.
**2016**, 123, 3–18. [Google Scholar] [CrossRef] - Gillespie, D.T. The mathematics of Brownian motion and Johnson noise. Am. J. Phys.
**1996**, 64, 225–240. [Google Scholar] [CrossRef] - Zili, M. On the mixed fractional brownian motion. J. Appl. Math. Stoch. Anal.
**2006**, 2006, 32435. [Google Scholar] [CrossRef] - Ben Adda, F.; Cresson, J. Corrigendum to “About non-differentiable functions”. J. Math. Anal. Appl.
**2013**, 408, 409–413. [Google Scholar] [CrossRef] - Kolwankar, K.M.; Lévy Véhel, J. Measuring functions smoothness with local fractional derivatives. Fract. Calc. Appl. Anal.
**2001**, 4, 285–301. [Google Scholar] - Samko, S.; Kilbas, A.; Marichev, O. Fractional Integrals and Derivatives: Theory and Applications; Gordon and Breach: Yverdon, Switzerland, 1993. [Google Scholar]
- De Rham, G. Sur quelques courbes definies par des equations fonctionnelles. Rendiconti del Seminario Matematico Università e Politecnico di Torino
**1957**, 16, 101–113. [Google Scholar] - Kolwankar, K.M.; Gangal, A.D. Fractional differentiability of nowhere differentiable functions and dimensions. Chaos
**1996**, 6, 505–513. [Google Scholar] [CrossRef] [PubMed] - Kolwankar, K.M.; Gangal, A.D. Local fractional Fokker-Planck equation. Phys. Rev. Lett.
**1998**, 80, 214–217. [Google Scholar] [CrossRef] - Tarasov, V.E. Local fractional derivatives of differentiable functions are integer-order derivatives or zero. Int. J. Appl. Comput. Math.
**2016**, 2, 195–201. [Google Scholar] [CrossRef] - Yang, X.J.; Baleanu, D.; Srivastava, H.M. Local Fractional Integral Transforms and Their Applications; Academic Press: Cambridge, MA, USA, 2015. [Google Scholar]
- Liu, Z.; Wang, T.; Gao, G. A local fractional Taylor expansion and its computation for insufficiently smooth functions. East Asian J. Appl. Math.
**2015**, 5, 176–191. [Google Scholar] [CrossRef] - Prodanov, D. Regularization of derivatives on non-differentiable points. J. Phys. Conf. Ser.
**2016**, 701, 012031. [Google Scholar] [CrossRef] - Prodanov, D. Fractional variation of Hölderian functions. Fract. Calc. Appl. Anal.
**2015**, 18, 580–602. [Google Scholar] [CrossRef]

**Figure 2.**Approximation of the fractional velocity of Neidinger’s function. Recursive construction of the fractional velocity for $\beta =1/3$ (

**top**) and $\beta =1/2$ (

**bottom**), iteration level 9. The Neidinger’s function IFS are given for comparison for the same iteration level.

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Prodanov, D.
Fractional Velocity as a Tool for the Study of Non-Linear Problems. *Fractal Fract.* **2018**, *2*, 4.
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**AMA Style**

Prodanov D.
Fractional Velocity as a Tool for the Study of Non-Linear Problems. *Fractal and Fractional*. 2018; 2(1):4.
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2018. "Fractional Velocity as a Tool for the Study of Non-Linear Problems" *Fractal and Fractional* 2, no. 1: 4.
https://doi.org/10.3390/fractalfract2010004