RELATIONSHIP BETWEEN UPPER BOX DIMENSION OF CONTINUOUS FUNCTIONS AND ORDERS OF WEYL FRACTIONAL INTEGRAL

Fractals ◽  
2021 ◽  
Author(s):  
H. B. GAO ◽  
Y. S. LIANG ◽  
W. XIAO
Keyword(s):  
Fractal Dimension ◽  
Continuous Function ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Fractional Integrals ◽  
Box Dimension ◽  
Hölder Continuous ◽  
Dimension One ◽  
Weyl Fractional Integral

In this paper, we mainly investigate relationship between fractal dimension of continuous functions and orders of Weyl fractional integrals. If a continuous function defined on a closed interval is of bounded variation, its Weyl fractional integral must still be a continuous function with bounded variation. Thus, both its Weyl fractional integral and itself have Box dimension one. If a continuous function satisfies Hölder condition, we give estimation of fractal dimension of its Weyl fractional integral. If a Hölder continuous function is equal to 0 on [Formula: see text], a better estimation of fractal dimension can be obtained. When a function is continuous on [Formula: see text] and its Weyl fractional integral is well defined, a general estimation of upper Box dimension of Weyl fractional integral of the function has been given which is strictly less than two. In the end, it has been proved that upper Box dimension of Weyl fractional integrals of continuous functions is no more than upper Box dimension of original functions.

COMPOSITION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS AND THEIR RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950065
Author(s):  
BIN YU ◽  
TAO ZHANG ◽  
LEI YAO ◽  
WEI ZHAO
Keyword(s):  
Fractal Dimension ◽  
Fractional Derivative ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Important Conclusion ◽  
One Dimensional ◽  
Dimension One

In this paper, we make research on composition of continuous functions with Box dimension one of bounded variation or unbounded variation on [Formula: see text]. It has been proved that one-dimensional continuous functions must be one of functions with bounded variation, or functions with finite unbounded variation points, or functions with infinite unbounded variation points on [Formula: see text]. Based on discussion of one-dimensional continuous functions, fractal dimension, such as Box dimension, of Riemann–Liouville (R-L) fractional integral of those functions have been calculated. We get an important conclusion that Box dimension of R-L fractional integral of any one-dimensional continuous functions of any positive orders still is one. R-L fractional derivative of certain one-dimensional continuous functions has been explored elementary.

THE EFFECT OF THE WEYL FRACTIONAL INTEGRAL ON FUNCTIONS

Fractals ◽  
2021 ◽  
Author(s):  
XIA TING ◽  
CHEN LEI ◽  
LUO LING ◽  
WANG YONG
Keyword(s):  
Linear Relationship ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Fractional Integrals ◽  
Hölder Condition ◽  
Weierstrass Function ◽  
Variation Function ◽  
Weyl Fractional Integral

This paper mainly discusses the influence of the Weyl fractional integrals on continuous functions and proves that the Weyl fractional integrals can retain good properties of many functions. For example, a bounded variation function is still a bounded variation function after the Weyl fractional integral. Continuous functions that satisfy the Holder condition after the Weyl fractional integral still satisfy the Holder condition, furthermore, there is a linear relationship between the order of the Holder conditions of the two functions. At the end of this paper, the classical Weierstrass function is used as an example to prove the above conclusion.

ESTIMATION OF FRACTAL DIMENSION OF WEYL FRACTIONAL INTEGRAL OF CERTAIN CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050030 ◽  
Author(s):  
YONG-SHUN LIANG
Keyword(s):  
Fractal Dimension ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Hölder Condition ◽  
Positive Order ◽  
Weyl Fractional Integral

In this work, we consider fractal dimension such as Box dimension, of Weyl fractional integral of certain continuous functions. Upper Box dimension of Weyl fractional integral of continuous functions satisfying [Formula: see text]-order Hölder condition of positive order [Formula: see text] is no more than both [Formula: see text] and [Formula: see text]. Furthermore, it is no more than [Formula: see text] which means strictly less than [Formula: see text]. Meanwhile, [Formula: see text], Box dimension of Weyl fractional integral of continuous functions satisfying [Formula: see text]-order Hölder condition must be one.

HÖLDER CONTINUITY AND BOX DIMENSION FOR THE WEYL FRACTIONAL INTEGRAL

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050032 ◽  
Author(s):  
LONG TIAN
Keyword(s):  
Continuous Function ◽  
Fractional Integral ◽  
Hölder Continuity ◽  
Box Dimension ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Weyl Fractional Integral

In this paper, we investigate the Hölder continuity and the estimate for the box dimension of the Weyl fractional integral of some continuous function [Formula: see text], denoted by [Formula: see text]. We obtain that if [Formula: see text] is [Formula: see text]-order Hölder continuous, then [Formula: see text] is [Formula: see text]-order Hölder continuous. Moreover, if [Formula: see text] belongs to [Formula: see text], then [Formula: see text] is [Formula: see text]-order Hölder continuous with [Formula: see text].

THE EFFECTS OF THE RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL ON THE BOX DIMENSION OF FRACTAL GRAPHS OF HÖLDER CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050052
Author(s):  
JUNRU WU
Keyword(s):  
Continuous Function ◽  
Fractional Integral ◽  
Closed Interval ◽  
Open Interval ◽  
Continuous Functions ◽  
Box Dimension ◽  
Hölder Continuous ◽  
Positive Order ◽  
Lipschitz Continuous ◽  
Hölder Continuous Functions

In this paper, the linearity of the dimensional-decrease effect of the Riemann–Liouville fractional integral is mainly explored. It is proved that if the Box dimension of the graph of an [Formula: see text]-Hölder continuous function is greater than one and the positive order [Formula: see text] of the Riemann–Liouville fractional integral satisfies [Formula: see text], the upper Box dimension of the Riemann–Liouville fractional integral of the graph of this function will not be greater than [Formula: see text]. Furthermore, it is proved that the Riemann–Liouville fractional integral of a Lipschitz continuous function defined on a closed interval is continuously differentiable on the corresponding open interval.

THE CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS AND THEIR FRACTIONAL INTEGRAL

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850063 ◽  
Author(s):  
XING LIU ◽  
JUN WANG ◽  
HE LIN LI
Keyword(s):  
Fractal Dimension ◽  
Continuous Function ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
One Dimensional

This paper mainly discusses the continuous functions whose fractal dimension is 1 on [Formula: see text]. First, we classify continuous functions into unbounded variation and bounded variation. Then we prove that the fractal dimension of both continuous functions of bounded variation and their fractional integral is 1. As for continuous functions of unbounded variation, we solve several special types. Finally, we give the example of one-dimensional continuous function of unbounded variation.

ESTIMATION OF FRACTAL DIMENSION OF FRACTIONAL CALCULUS OF THE HÖLDER CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050123
Author(s):  
YONG-SHUN LIANG
Keyword(s):  
Fractal Dimension ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Hölder Condition ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Liouville Fractional Derivative

In the present paper, fractal dimension and properties of fractional calculus of certain continuous functions have been investigated. Upper Box dimension of the Riemann–Liouville fractional integral of continuous functions satisfying the Hölder condition of certain positive orders has been proved to be decreasing linearly. If sum of order of the Riemann–Liouville fractional integral and the Hölder condition equals to one, the Riemann–Liouville fractional integral of the function will be Lipschitz continuous. If the corresponding sum is strictly larger than one, the Riemann–Liouville fractional integral of the function is differentiable. Estimation of fractal dimension of the derivative function has also been discussed. Finally, the Riemann–Liouville fractional derivative of continuous functions satisfying the Hölder condition exists when order of the Riemann–Liouville fractional derivative is smaller than order of the Hölder condition. Upper Box dimension of the function has been proved to be increasing at most linearly.

BOX DIMENSION OF HADAMARD FRACTIONAL INTEGRAL OF CONTINUOUS FUNCTIONS OF BOUNDED AND UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (03) ◽  
pp. 1750035 ◽  
Author(s):  
XIAO ER WU ◽  
JUN HUAI DU
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
Box Dimension ◽  
Definition Of ◽  
Hadamard Fractional Integral ◽  
Variation Point

The present paper investigates fractal dimension of Hadamard fractional integral of continuous functions of bounded and unbounded variation. It has been proved that Hadamard fractional integral of continuous functions of bounded variation still is continuous functions of bounded variation. Definition of an unbounded variation point has been given. We have proved that Box dimension and Hausdorff dimension of Hadamard fractional integral of continuous functions of bounded variation are [Formula: see text]. In the end, Box dimension and Hausdorff dimension of Hadamard fractional integral of certain continuous functions of unbounded variation have also been proved to be [Formula: see text].

SOME REMARKS ON FRACTIONAL INTEGRAL OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (01) ◽  
pp. 2050005
Author(s):  
JIA YAO ◽  
YING CHEN ◽  
JUNQIAO LI ◽  
BIN WANG
Keyword(s):  
Continuous Function ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Positive Order ◽  
One Dimensional ◽  
Katugampola Fractional Integral ◽  
Hadamard Fractional Integral

In this paper, we make research on Katugampola and Hadamard fractional integral of one-dimensional continuous functions on [Formula: see text]. We proved that Katugampola fractional integral of bounded and continuous function still is bounded and continuous. Box dimension of any positive order Hadamard fractional integral of one-dimensional continuous functions is one.

Sign in / Sign up

Export Citation Format

Share Document