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.

FRACTAL DIMENSIONS OF WEYL–MARCHAUD FRACTIONAL DERIVATIVE OF CERTAIN ONE-DIMENSIONAL FUNCTIONS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950114
Author(s):  
Y. S. LIANG ◽  
N. LIU
Keyword(s):  
Fractional Derivative ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
Box Dimension ◽  
One Dimensional ◽  
Dimension One

Fractal dimensions of Weyl–Marchaud fractional derivative of certain continuous functions are investigated in this paper. Upper Box dimension of Weyl–Marchaud fractional derivative of certain continuous functions with Box dimension one has been proved to be no more than the sum of one and its order.

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.

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.

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.

FRACTAL DIMENSIONS OF LINEAR COMBINATION OF CONTINUOUS FUNCTIONS WITH THE SAME BOX DIMENSION

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050139
Author(s):  
XUEFEI Wang ◽  
CHUNXIA ZHAO
Keyword(s):  
Linear Combination ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
Box Dimension ◽  
Fractal Function ◽  
Space Of Continuous Functions ◽  
Fractal Functions

In this paper, we mainly discuss continuous functions with certain fractal dimensions on [Formula: see text]. We find space of continuous functions with certain Box dimension is not closed. Furthermore, Box dimension of linear combination of two continuous functions with the same Box dimension maybe does not exist. Definitions of fractal functions and local fractal functions have been given. Linear combination of a fractal function and a local fractal function with the same Box dimension must still be the original Box dimension with nontrivial coefficients.

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.

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.

DIMENSION ANALYSIS OF CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1730001 ◽  
Author(s):  
JUN WANG ◽  
KUI YAO
Keyword(s):  
Hausdorff Dimension ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
Packing Dimension ◽  
Box Dimension ◽  
Box Counting ◽  
Dimension Analysis ◽  
Box Counting Dimension ◽  
Self Similar

In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are [Formula: see text]. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is [Formula: see text] also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be [Formula: see text] when [Formula: see text] is self-similar.

THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS

Fractals ◽  
1995 ◽  
Vol 03 (01) ◽  
pp. 217-229 ◽  
Author(s):  
FRANK B. TATOM
Keyword(s):  
Time Series ◽  
Fractal Dimension ◽  
Fractional Calculus ◽  
Fractal Dimensions ◽  
Linear Functions ◽  
Fractal Curve ◽  
Random Fractal ◽  
Fractal Functions ◽  
Decay Exponent ◽  
The Relationship

The general relationship between fractional calculus and fractals is explored. Based on prior investigations dealing with random fractal processes, the fractal dimension of the function is shown to be a linear function of the order of fractional integro-differentiation. Emphasis is placed on the proper application of fractional calculus to the function of the random fractal, as opposed to the trail. For fractional Brownian motion, the basic relations between the spectral decay exponent, Hurst exponent, fractal dimension of the function and the trail, and the order of the fractional integro-differentiation are developed. Based on an understanding of fractional calculus applied to random fractal functions, consideration is given to an analogous application to deterministic or nonrandom fractals. The concept of expressing each coordinate of a deterministic fractal curve as a “pseudo-time” series is investigated. Fractional integro-differentiation of such series is numerically carried out for the case of quadric Koch curves. The resulting time series produces self-similar patterns with fractal dimensions which are linear functions of the order of the fractional integro-differentiation. These curves are assigned the name, fractional Koch curves. The general conclusion is reached that fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates which can be expressed as functions of one independent variable, which is typically time (or pseudo-time).

scholarly journals ON SOME QUALITATIVE PROPERTIES OF THE OPERATOR OF FRACTIONAL DIFFERENTIATION IN KIPRIYANOV SENSE

2017 ◽  
Vol 23 (2) ◽  
pp. 32-43
Author(s):  
M. V. Kukushkin
Keyword(s):  
Fractional Calculus ◽  
Fractional Integral ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Fractional Differentiation ◽  
Qualitative Properties ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Integral Equality ◽  
Definition Of

In this paper we investigated the qualitative properties of the operator of fractional differentiation in Kipriyanov sense. Based on the concept of multidimensional generalization of operator of fractional differentiation in Marchaud sense we have adapted earlier known techniques of proof theorems of one-dimensional theory of fractional calculus for the operator of fractional differentiation in Kipriyanov sense. Along with the previously known definition of the fractional derivative in the direction we used a new definition of multidimensional fractional integral in the direction of allowing you to expand the domain of definition of formally adjoint operator. A number of theorems that have analogs in one-dimensional theory of fractional calculus is proved. In particular the sufficient conditions of representability of a fractional integral in the direction are received. Integral equality the result of which is the construction of the formal adjoint operator defined on the set of functions representable by the fractional integral in direction is proved.

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