The relationship between the order of (k, s)-Riemann-Liouville fractional integral and the fractal dimensions of a fractal function

Navish, A. A.; Priya, M.; Uthayakumar, R.

doi:10.1007/s41478-022-00451-9

Cited by 4 publications

(1 citation statement)

References 12 publications

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“…Recently, Priya and Uthayakumar [33] observed that the Hausdorff dimension and the Box dimension of the graph of a continuous function under (k, s)-RLFI are both one. Moreover, the linear relationship between the fractal dimension of (k, s)-RLFI of the Weierstrass functions and the fractional order has been discussed in [34].…”

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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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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On the variable order Weyl-Marchaud fractional derivative of non-affine fractal function

Chinnathambi

Gowrisankar

2023

J Anal

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FRACTAL DIMENSIONS FOR THE MIXED (κ,s)-RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL OF BIVARIATE FUNCTIONS

WANG,

XIAO

2024

Fractals

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The research object of this paper is the mixed [Formula: see text]-Riemann–Liouville fractional integral of bivariate functions on rectangular regions, which is a natural generalization of the fractional integral of univariate functions. This paper first indicates that the mixed integral still maintains the validity of the classical properties, such as boundedness, continuity and bounded variation. Furthermore, we investigate fractal dimensions of bivariate functions under the mixed integral, including the Hausdorff dimension and the Box dimension. The main results indicate that fractal dimensions of the graph of the mixed [Formula: see text]-Riemann–Liouville integral of continuous functions with bounded variation are still two. The Box dimension of the mixed integral of two-dimensional continuous functions has also been calculated. Besides, we prove that the upper bound of the Box dimension of bivariate continuous functions under [Formula: see text] order of the mixed integral is [Formula: see text] where [Formula: see text].

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The relationship between the order of (k, s)-Riemann-Liouville fractional integral and the fractal dimensions of a fractal function

Cited by 4 publications

References 12 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

On the variable order Weyl-Marchaud fractional derivative of non-affine fractal function

FRACTAL DIMENSIONS FOR THE MIXED (κ,s)-RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL OF BIVARIATE FUNCTIONS

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