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.

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.

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.

FRACTAL DIMENSION OF RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL OF CERTAIN UNBOUNDED VARIATIONAL CONTINUOUS FUNCTION

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750047 ◽  
Author(s):  
YANG LI ◽  
WEI XIAO
Keyword(s):  
Fractal Dimension ◽  
Continuous Function ◽  
Fractional Integral ◽  
Box Dimension ◽  
One Dimensional

In the present paper, a one-dimensional continuous function of unbounded variation on the interval [Formula: see text] has been constructed. Box dimension of this function has been proved to be 1. Furthermore, Box dimension of its Riemann–Liouville fractional integral of any order has also been proved to be 1.

Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions

2018 ◽  
Vol 21 (6) ◽  
pp. 1651-1658 ◽  
Author(s):  
Yong Shun Liang
Keyword(s):  
Fractal Dimension ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Positive Order

Abstract The present paper investigates fractal dimension of fractional integral of continuous functions whose fractal dimension is 1 on [0, 1]. For any continuous functions whose Box dimension is 1 on [0, 1], Riemann-Liouville fractional integral of these functions of any positive order has been proved to still be 1-dimensional continuous functions on [0, 1].

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 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.

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.

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].

PROGRESS ON ESTIMATION OF FRACTAL DIMENSIONS OF FRACTIONAL CALCULUS OF CONTINUOUS FUNCTIONS

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950084 ◽  
Author(s):  
YONG-SHUN LIANG
Keyword(s):  
Fractional Calculus ◽  
Fractional Integral ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
Box Dimension ◽  
One Dimensional ◽  
Zero Measure ◽  
Liouville Fractional Derivative ◽  
Fractal Functions ◽  
Dimension One

In this paper, fractal dimensions of fractional calculus of continuous functions defined on [Formula: see text] have been explored. Continuous functions with Box dimension one have been divided into five categories. They are continuous functions with bounded variation, continuous functions with at most finite unbounded variation points, one-dimensional continuous functions with infinite but countable unbounded variation points, one-dimensional continuous functions with uncountable but zero measure unbounded variation points and one-dimensional continuous functions with uncountable and non-zero measure unbounded variation points. Box dimension of Riemann–Liouville fractional integral of any one-dimensional continuous functions has been proved to be with Box dimension one. Continuous functions on [Formula: see text] are divided as local fractal functions and fractal functions. According to local structure and fractal dimensions, fractal functions are composed of regular fractal functions, irregular fractal functions and singular fractal functions. Based on previous work, upper Box dimension of any continuous functions has been proved to be no less than upper Box dimension of their Riemann–Liouville fractional integral. Fractal dimensions of Riemann–Liouville fractional derivative of certain continuous functions have been investigated elementary.

THE RELATIONSHIP BETWEEN FRACTAL DIMENSIONS OF BESICOVITCH FUNCTION AND THE ORDER OF HADAMARD FRACTIONAL INTEGRAL

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050128
Author(s):  
BIN WANG ◽  
WENLONG JI ◽  
LEGUI ZHANG ◽  
XUAN LI
Keyword(s):  
Fractional Integral ◽  
Fractal Dimensions ◽  
Packing Dimension ◽  
Box Dimension ◽  
Dimension Formula ◽  
The Relationship ◽  
Hadamard Fractional Integral

In this paper, we mainly research on Hadamard fractional integral of Besicovitch function. A series of propositions of Hadamard fractional integral of [Formula: see text] have been proved first. Then, we give some fractal dimensions of Hadamard fractional integral of Besicovitch function including Box dimension, [Formula: see text]-dimension and Packing dimension. Finally, relationship between the order of Hadamard fractional integral and fractal dimensions of Besicovitch function has also been given.

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