Diffusion on Fractal Ces\`aro Curve

TL;DR

This paper extends calculus to fractal Koch and Cesàro curves, proposing a generalized Newton's law and analyzing particle density on these fractals with illustrative examples.

Contribution

It introduces F-calculus on fractal curves and formulates a generalized Newton's second law specific to these fractals, providing new tools for fractal dynamics analysis.

Findings

Derived particle density on fractal Cesàro curves.

Presented detailed examples of F-integrals and F-derivatives.

Proposed a generalized Newton's second law for fractal curves.

Abstract

In this paper, we apply F-calculus on fractal Koch and Ces\`aro curves with different dimensions. Generalized Newton's second law on the fractal Koch and Ces\`aro curves is suggested. Density of moving particles which absorbed on fractal Ces\`aro are derived. More, illustrative examples are given to present the details of F-integrals and F-derivatives.

Equations

$${\gamma }^{\alpha }(F,a,b)=\underset{\delta \to 0}{\lim }\underset{\{P_{[a,b]}:|P|\le \delta \}}{\inf }\sum _{i=0}^{n-1}\frac{|\text{w}(t_{i+1})-\text{w}(t_i){|}^{\alpha }}{\Gamma (\alpha +1)},$$

$$S_F^{\alpha }(t)=\{\begin{array}{l}{\gamma }^{\alpha }(F,p_0,t)t\ge p_0,\\ -{\gamma }^{\alpha }(F,t,p_0)t<p_0,\end{array}$$

$${\dim }_{\gamma }(F)=\inf \{\alpha :{\gamma }^{\alpha }(F,a,b)=0\}$$

$$=\sup \{\alpha :{\gamma }^{\alpha }(F,a,b)=\infty \}.$$

$$D_F^{\alpha }f(\theta )=F-\underset{{\theta }^{\prime }\to \theta }{\lim }\frac{f({\theta }^{\prime })-f(\theta )}{J({\theta }^{\prime })-J(\theta )},$$

$${\theta }^{\prime }\in F\text{and}|{\theta }^{\prime }-\theta |<\delta \Rightarrow |f({\theta }^{\prime })-l|<ϵ.$$

$$l=F-\underset{{\theta }^{\prime }\to \theta }{\lim }f({\theta }^{\prime }).$$

$$C(t_1,t_2)=\{w(t^{\prime }):t^{\prime }\in [t_1,t_2]\},$$

$$M[f,C(t_1,t_2)]=\underset{\theta \in C(t_1,t_2)}{\sup }f(\theta ),$$

$$m[f,C(t_1,t_2)]=\underset{\theta \in C(t_1,t_2)}{\inf }f(\theta ).$$

$$U^{\alpha }[f,F,P]=\sum _{i=0}^{n-1}M[f,C(t_i,t_{i+1})][S_F^{\alpha }(t_{i+1})-S_F^{\alpha }(t_i)],$$

$$L^{\alpha }[f,F,P]=\sum _{i=0}^{n-1}m[f,C(t_i,t_{i+1})][S_F^{\alpha }(t_{i+1})-S_F^{\alpha }(t_i)].$$

$${\int }_{C(a,b)}f(\theta )d_F^{\alpha }\theta =\underline{{\int }_{C(a,b)}}f(\theta )d_F^{\alpha }\theta =\underset{P_{[a,b]}}{\sup }{L}^{\alpha }[f,F,P]$$

$$=\overline{{\int }_{C(a,b)}}f(\theta )d_F^{\alpha }\theta =\underset{P_{[a,b]}}{\inf }{U}^{\alpha }[f,F,P].$$

$${\int }_{C(a,b)}h(\theta )d_F^{\alpha }\theta =f(w(b))-f(w(a)).$$

$$g(w(t))={\int }_{C(a,t)}f(\theta )d_F^{\alpha }\theta ,t\in [a,b],$$

$$D_F^{\alpha }g(\theta )=f(\theta ).$$

$$h(\theta )=\sum _{n=0}^{\infty }\frac{(J(\theta )-J({\theta }^{\prime }){)}^n}{n!}(D_F^{\alpha }{)}^{n}h({\theta }^{\prime }),\theta \in F.$$

$${\int }_{C(a,b)}f(\theta )d_F^{\alpha }\theta ={\int }_{C(a,b)}1d_F^{\alpha }\theta$$

$$=S_F^{\alpha }(b)-S_F^{\alpha }(a)=J((w(b))-J((w(a)),$$

$$f(t)=(S_F^{\alpha }(t){)}^2.$$

$${\int }_{C(0,t)}f(t)d_F^{\alpha }t=\frac{(S_F^{\alpha }(t){)}^3}{3}+k,$$

$$D_F^{\alpha }f(t)=2S_F^{\alpha }(t),$$

$$m(D_F^{\alpha }{)}^2{\text{r}}_F^{\alpha }(t)={\text{f}}_F^{\alpha },$$

$${\text{a}}_F^{\alpha }(t)$$

$${\text{v}}_F^{\alpha }(t)$$

$${\text{r}}_F^{\alpha }(t)$$

$$D_F^{\beta }\zeta (t)=-k\zeta (t),\beta =\mathrm{1.78.}$$

$$\zeta (t)=\zeta (0)e^{-k{S}_F^{\beta }(t)}.$$

Full text

Keywords: $F^{\alpha}$-calculus; Fractal Koch curve; Staircase function; Fractal Cesàro curve

a Young Researchers and Elite Club, Urmia Branch, Islamic Azad University, Urmia, Iran

†E-mail address: alirezakhalili2002@yahoo.co.in

1 Introduction

Fractional calculus which include derivatives and integrals with arbitrary orders is applied in science, engineering and etc. [1, 2, 3]. The fractional derivatives are used to model non-conservative systems, processes with the memory effect and anomalous diffusion [4, 5, 6]. The fractional derivatives are non-local but the most measurements in physics are local [7]. As a result, in view of the fractional local derivatives and Chapman-Kolmogorov condition new Fokker-Planck equation is given [8]. The local fractional derivatives lead to new measure on fractal sets [9]. Fractal geometry which is the generalized Euclidean geometry has important role in science, engineering and medical science. Fractals are the shapes which have self-similar properties and fractional dimensions. Many methods have been used to build analysis on fractal sets and processes [10, 11, 12, 13, 14, 15, 16].

Recently, $F^{\alpha}$-calculus is suggested in the seminal paper as a framework by A. Parvate and A. D. Gangal which is the generalized standard calculus. $F^{\alpha}$-calculus is calculus on fractals with the algorithmic property [17, 18, 19]. Researchers have explored this area giving new insight into $F^{\alpha}$-calculus [20, 21, 22, 23]. The fractal Cantor sets are considered as grating in the diffraction phenomena [24]. Regarding the above mentioned research we apply the $F^{\alpha}$-calculus on fractal curves in the case of fractal Koch curves. Differential equation corresponding for a motion of the particle on fractal curves is purposed.

Outline of the paper is as follows:

In Section 2 we summarize $F^{\alpha}$-calculus on fractal Koch and Cesàro curves without proofs. In Section 3, we give our new result in this paper which includes equation of motion of the particles. Section 3 contains our conclusion.

2 Preliminaries

In this section, we summarize $F^{\alpha}$-calculus on parameterize fractal curves and use in the case of fractal Koch and Cesàro curves (see for review Refs. [19, 25].

**Calculus on fractal Koch and Cesàro curves:

**Let us consider fractal Koch and Cesàro curves which are denoted by $F\subset\Re^{3}$ and define corresponding staircase function. Fractal Koch and Cesàro curves are called continuously parameterizable if there exists a function $\textbf{w}(t):[a_{0},b_{0}]\rightarrow F,~{}a_{0},b_{0}\in\Re$ which is continuous one to one and onto $F$ [19, 25].

Definition : For the fractal curves $F$ and a subdivision $P_{[a,b]},a<b,[a,b]\subset[a_{0},b_{0}]$ mass function is defined [19]

[TABLE]

where $|.|$ indicates the Euclidean norm on $R^{3}$ and $P_{[a,b]}=\{a=t_{0},...,t_{n}=b\}.$

Definition: The staircase functions for fractal Koch and Cesàro curves are defined

[TABLE]

where $p_{0}\in[a_{0},b_{0}]$ is arbitrary point.

In Figure [1] we have sketched fractal Koch and Cesáro curves and $S_{F}^{\alpha}(t)$ setting $\alpha=1.26$ and $\alpha=1.78$.

Definition: The $\gamma$-dimension of fractal Koch and Cesáro curves ($F$) are defined

[TABLE]

Definition: $F^{\alpha}$-derivative of function $f$ at $\theta\in F$ is defined

[TABLE]

where $J(\theta)=S_{F}^{\alpha}(w^{-1}(\theta)),\theta\in F$ and if the limit exists [19, 25].

**Definition: ** A number $l$ is $F$-limit of the function $f$ if we have

[TABLE]

If such a number exists [19]. It is indicate by

[TABLE]

A segment $C(t_{1},t_{2})$ of fractal Koch and Cesáro curve is define as

[TABLE]

and $M,~{}m$ are defined as follows [19, 25]

[TABLE]

[TABLE]

**Definition:  **The upper and the lower $F^{\alpha}$-sum for the function $f$ over the subdivision $P$ are defined

[TABLE]

[TABLE]

Definition: $F^{\alpha}$-integral of the function $f$ is defined

[TABLE]

Fundamental theorems of $F^{\alpha}$-calculus:

First Part: If $f:F\rightarrow R$ is $F^{\alpha}$-differentiable function and $h:F\rightarrow R$ is $F$-continuous such that $h(\theta)=D_{F}^{\alpha}f(\theta)$, then we have

[TABLE]

Second part: If $f$ is bounded, $F$-continuous on $C(a,b)$ and $g:F\rightarrow R$ then we have

[TABLE]

where we have

[TABLE]

For the proofs we refer the reader to [19].

**Some of the properties:

**1) If $f(\theta)=k\in R$ then we have $D_{F}^{\alpha}f=0$ .

2. If $f$ is a $F$-continuous and $D_{F}^{\alpha}f=0$ then $f=k$.

3. Generalized Taylor Series on fractal Koch curves is

[TABLE]

[TABLE]

where $f(\theta)=1$ is constant function [19, 25].

Note: $F^{\alpha}$-derivative and $F^{\alpha}$-integral on fractal Koch curves are linear operators.

Example 1. Consider $f(t):F\rightarrow R$ on the fractal Koch curves as

[TABLE]

The $F^{\alpha}$-derivative and the $F^{\alpha}$-integral of $f$ are

[TABLE]

and

[TABLE]

where $k$ is constant. Figure [2] shows the graphs of $f$, $F^{\alpha}$-integral of $f$, and $F^{\alpha}$-derivative of $f$.

3 Equation of motion on fractal curves

Generalized Newton’s second law on fractal Koch and Cesáro curves is suggested

[TABLE]

where $\textbf{r}_{F}^{\alpha}:F\rightarrow\Re$,  $\textbf{v}_{F}^{\alpha}(t)=D_{F}^{\alpha}~{}\textbf{r}_{F}^{\alpha}$ and $\textbf{a}_{F}^{\alpha}(t)=(D_{F}^{\alpha})^{2}~{}\textbf{r}_{F}^{\alpha}$ are called generalized position, generalized velocity and generalized acceleration on fractal Koch and Cesáro curves, respectively.

Example 2. Consider a force $\textbf{f}^{\alpha}_{F}=k(\hat{i}+\hat{j})~{}[~{}ML^{\alpha}T^{2}~{}]$ such that $\textbf{f}^{\alpha}_{F}:F\rightarrow\Re$ apply on a particle with mass $m$ on fractal Koch curves. One sees immediately that generalized acceleration, velocity and position are

[TABLE]

We preset the graph of the Eq. (20) in Figure [3]

Example 3. Consider particles moving along the fractal Cesàro curve which absorb the particles. The mathematical model for this phenomenon is given by

[TABLE]

where $\zeta$ is the density of particles on fractal Cesàro curve. Using $F^{\alpha}$-integral, it is easy to obtain the solution

[TABLE]

Figure [4] shows the graph of $\zeta(t)$ on fractal Cesàro curve.

4 Conclusion

In this paper, $F^{\alpha}$-calculus is the generalization of the standard calculus on the fractals with fractional dimension and self-similar properties. In the sense of the standard calculus the fractal Koch and Cesàro curves are not differentiable and integrable. The $F^{\alpha}$-calculus is used to define $F^{\alpha}$-integral and $F^{\alpha}$-derivative on fractal Koch and Cesàro curves. Some illustrative examples are given for presenting the details. Finally, generalized differential equation corresponding to the motions on the fractal Koch and Cesàro curves are suggested and solved.