Further Discussion on Relationship between Fractal Dimension of Weierstrass Function and Order of Riemann-Liouville Fractional Integral

2020 ◽  
Vol 10 (11) ◽  
pp. 1035-1043
Author(s):  
鸿博 高
Keyword(s):  
Fractal Dimension ◽  
Fractional Integral ◽  
Weierstrass Function

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.

FRACTAL DIMENSION OF A SPECIAL CONTINUOUS FUNCTION

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850048 ◽  
Author(s):  
NING LIU ◽  
KUI YAO
Keyword(s):  
Fractal Dimension ◽  
Continuous Function ◽  
Hausdorff Dimension ◽  
Fractional Integral ◽  
Fractal Function

In this paper, we mainly construct a special fractal function defined on [Formula: see text] of unbounded variation by method of iteration, and prove that both Box and Hausdorff dimension of this function be 1. Further, we discuss the Riemann–Liouville fractional integral of this function. Finally, some numerical and graphic results are provided to characterize this special fractal function.

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].

New relationships connecting a class of fractal objects and fractional integrals in space

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Raoul Nigmatullin ◽  
Dumitru Baleanu
Keyword(s):  
Fractal Dimension ◽  
Fractional Integral ◽  
The Self ◽  
Fractional Integrals ◽  
Smooth Functions ◽  
New Approach ◽  
Power Law Exponent ◽  
Microscopic Function ◽  
Self Similar ◽  
The Given

AbstractMany specialists working in the field of the fractional calculus and its applications simply replace the integer differentiation and integration operators by their non-integer generalizations and do not give any serious justifications for this replacement. What kind of “Physics” lies in this mathematical replacement? Is it possible to justify this replacement or not for the given type of fractal and find the proper physical meaning? These or other similar questions are not discussed properly in the current papers related to this subject. In this paper new approach that relates to the procedure of the averaging of smooth functions on a fractal set with fractional integrals is suggested. This approach contains the previous one as a partial case and gives new solutions when the microscopic function entering into the structural-factor does not have finite value at N ≫ 1 (N is number of self-similar objects). The approach was tested on the spatial Cantor set having M bars with different symmetry. There are cases when the averaging procedure leads to the power-law exponent that does not coincide with the fractal dimension of the self-similar object averaged. These new results will help researches to understand more clearly the meaning of the fractional integral. The limits of applicability of this approach and class of fractal are specified.

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