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

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.

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

Fractals ◽

10.1142/s0218348x19500841 ◽

2019 ◽

Vol 27 (05) ◽

pp. 1950084 ◽

Cited By ~ 2

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.

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SOME REMARKS ON FRACTIONAL INTEGRAL OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS

Fractals ◽

10.1142/s0218348x2050005x ◽

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.

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DIMENSION ANALYSIS OF CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽

10.1142/s0218348x1730001x ◽

2017 ◽

Vol 25 (01) ◽

pp. 1730001 ◽

Cited By ~ 17

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.

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

Fractals ◽

10.1142/s0218348x2050139x ◽

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.

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FRACTAL DIMENSION OF CERTAIN CONTINUOUS FUNCTIONS OF UNBOUNDED VARIATION

Fractals ◽

10.1142/s0218348x17500098 ◽

2017 ◽

Vol 25 (01) ◽

pp. 1750009 ◽

Cited By ~ 13

Author(s):

Y. S. LIANG ◽

W. Y. SU

Keyword(s):

Fractal Dimension ◽

Continuous Function ◽

Bounded Variation ◽

Closed Interval ◽

Fractal Dimensions ◽

Continuous Functions ◽

One Dimensional ◽

Closed Intervals ◽

Bounded Variation Functions

Continuous functions on closed intervals are composed of bounded variation functions and unbounded variation functions. Fractal dimension of continuous functions with bounded variation must be one-dimensional (1D). While fractal dimension of continuous functions with unbounded variation may be 1 or not. Certain continuous functions of unbounded variation whose fractal dimensions are 1 have been mainly investigated in the paper. A continuous function on a closed interval with finite unbounded variation points has been proved to be 1D. Furthermore, we deal with continuous functions which have infinite unbounded variation points and part of them have been proved to be 1D. Certain examples of 1D continuous functions which have uncountable unbounded variation points have been given in the present paper.

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

Fractals ◽

10.1142/s0218348x20501236 ◽

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.

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

Fractals ◽

10.1142/s0218348x20502230 ◽

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.

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On Finite Part Integrals and Hadamard-Type Fractional Derivatives

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4037930 ◽

2018 ◽

Vol 13 (9) ◽

Cited By ~ 8

Author(s):

Li Ma ◽

Changpin Li

Keyword(s):

Fractional Derivative ◽

Singular Integral ◽

Fractional Derivatives ◽

Continuous Functions ◽

Finite Part ◽

Absolutely Continuous Functions ◽

Absolutely Continuous ◽

Fundamental Properties ◽

Logarithmic Series ◽

The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

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Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

Advances in Difference Equations ◽

10.1186/s13662-021-03472-z ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Raheel Kamal ◽

Kamran ◽

Gul Rahmat ◽

Ali Ahmadian ◽

Noreen Izza Arshad ◽

...

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Laplace Transform ◽

Fractional Derivative ◽

Meshless Method ◽

Inverse Laplace Transform ◽

Contour Integral ◽

Partial Differential ◽

One Dimensional ◽

The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

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