The relationship between the fractal dimensions of a type of fractal functions and the order of their fractional calculus

Abstract

The present paper investigate the relationship between the Hausdorff dimension of a type of fractal functions and the order of their Riemann–Liouville fractional calculus. Two examples of obvious practical value have been shown.

Introduction

A variety of interesting fractals, both of theoretical and practical importance, occur as graphs of functions. Indeed, many phenomena display fractal features when plotted as functions of time. Examples include atmospheric pressure, levels of reservoirs and prices on the stock market, at least when recorded over fairly long time spans. Fractal functions are also fundamental to E-infinity theory (see Refs. [12], [13], [14], [15], etc). On one hand, graphs of such functions are special fractals. A lot of fractals of E-infinity theory are composed of such fractals. On the other hand, they are special functions. We can make fractional calculus on them. So it will be important complement of E-infinity theory.

It is important and necessary to discuss the variance ratio of such functions. But they are often nowhere differentiable, even they are continuous function. So, we appeal to fractional calculus which used to investigate fractal functions as a tool becomes more and more important in the studies of fractal analysis (see Refs. [1], [3], [4], [8], [9], [10], [11], etc).

Dimension is the most important topological invariant of a topology. It is more fundamental and basic than the Euler class. In Ref. [8], Yao proved that there exist some linear connection between the order of the fractional calculus and the fractal dimensions of the graph of Weierstrass function. He mainly considered the Box, Packing and K-dimension. However, the Hausdorff dimension is more accurate than the three dimensions. Our present paper considers the relationship between the Hausdorff dimension of the Besicovitch functions and the order of their Riemann–Liouville fractional calculus.

The Riemann–Liouville fractional calculus is defined as follows:

Definition 1.1 [5], [6]

Let f be piecewisely continuous on $(0\text{,}\infty )$, and local integrable on $[0\text{,}\infty )$. Then for $t>0\text{,}\mathit{Re}(v)>0$, we call

$$D^{-v}f(t)=\frac{1}{\Gamma (v)}{\int }_0^t(t-x{)}^{v-1}f(x)\mathrm{d}x$$

the Riemann–Liouville fractional integral of f of order v. For $0<u<1$, we call

$$D^{u}f(t)=D[D^{u-1}f(t)]$$

the Riemann–Liouville fractional derivative of f of order u.

Now we are mainly interested in considering some generalized fractal functions, i.e., the Besicovitch functions which is defined by

$$B(t)=\sum _{n⩾1}{\lambda }_n^{-\alpha }\sin ({\lambda }_{n}t)\text{,}0<\alpha <1\text{,}{\lambda }_n\nearrow +\infty \text{.}$$

For $0<\alpha \text{,}v\text{,}u<1$ with $0<\alpha +v<1\text{,}0<u<\alpha \text{,}{\lambda }_1>1$, denote by

$$g(t)≔D^{-v}B(t)=\sum _{n⩾1}{\lambda }_n^{-\alpha }{S}_t(v\text{,}{\lambda }_n)$$

the fractional integral of Besicovitch functions $B(t)$ of order v, and denote by

$$m(t)≔D^{u}B(t)=\sum _{n⩾1}{\lambda }_n^{1-\alpha }{C}_t(1-u\text{,}{\lambda }_n)$$

the fractional derivative of Besicovitch functions $B(t)$ of order u. Here $S_t(v\text{,}a)\text{,}{C}_t(v\text{,}a)$ denote the fractional calculus of $\sin \mathit{at}$ and $\cos \mathit{at}$ of order v respectively.