Approximation with continuous functions preserving fractal dimensions of the Riemann-Liouville operators of fractional calculus

Yu, Binyan; Liang, Yongshun

doi:10.1007/s13540-023-00215-7

Cited by 13 publications

(2 citation statements)

References 44 publications

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“…[17][18][19][20] for more details. More recent work about the fractal dimensions of the graph of continuous functions can be found in [21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral

Wang,

Xiao

2023

Symmetry

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This article is a study on the (k,s)-Riemann–Liouville fractional integral, a generalization of the Riemann–Liouville fractional integral. Firstly, we introduce several properties of the extended integral of continuous functions. Furthermore, we make the estimation of the Box dimension of the graph of continuous functions after the extended integral. It is shown that the upper Box dimension of the (k,s)-Riemann–Liouville fractional integral for any continuous functions is no more than the upper Box dimension of the functions on the unit interval I=[0,1], which indicates that the upper Box dimension of the integrand f(x) will not be increased by the σ-order (k,s)-Riemann–Liouville fractional integral ksD−σf(x) where σ>0 on I. Additionally, we prove that the fractal dimension of ksD−σf(x) of one-dimensional continuous functions f(x) is still one.

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“…[17][18][19][20] for more details. More recent work about the fractal dimensions of the graph of continuous functions can be found in [21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral

Wang,

Xiao

2023

Symmetry

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“…This problem can be traced back to the research made first by Falconer [33], who showed that the box dimension of the sum of two continuous functions equals the greater of the box dimensions of them. On this basis, a group of academic workers has pushed this study forward and obtained a series of preliminary conclusions, whose related progress can be found in [34][35][36][37][38][39][40]. So in this paper, we shall focus on the fractal dimension of the superposition of two fractal surfaces and investigate whether it has the same result as that of fractal curves.…”

Section: Introductionmentioning

confidence: 99%

An Investigation on Fractal Characteristics of the Superposition of Fractal Surfaces

Wang

2023

Fractal Fract

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In this paper, we conduct research on the fractal characteristics of the superposition of fractal surfaces from the view of fractal dimension. We give the upper bound of the lower and upper box dimensions of the graph of the sum of two bivariate continuous functions and calculate the exact values of them under some particular conditions. Further, it has been proven that the superposition of two continuous surfaces cannot keep the fractal dimensions invariable unless both of them are two-dimensional. A concrete example of a numerical experiment has been provided to verify our theoretical results. This study can be applied to the fractal analysis of metal fracture surfaces or computer image surfaces.

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On two special classes of fractal surfaces with certain Hausdorff and Box dimensions

Yu,

Liang

2024

Applied Mathematics and Computation

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Approximation with continuous functions preserving fractal dimensions of the Riemann-Liouville operators of fractional calculus

Cited by 13 publications

References 44 publications

The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral

The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral

An Investigation on Fractal Characteristics of the Superposition of Fractal Surfaces

On two special classes of fractal surfaces with certain Hausdorff and Box dimensions

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