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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (20 November 2021) | Viewed by 20783

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Special Issue Editors

Prof. Dr. María A. Navascués

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Guest Editor

Department of Applied Mathematics, Universidad de Zaragoza, 50018 Zaragoza, Spain
Interests: fractals; fixed point theory; approximation; iterative methods
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. María Victoria Sebastián

E-Mail Website
Guest Editor

Centro Universitario de la Defensa-Academia General Militar, 50090 Zaragoza, Spain
Interests: fractal interpolation functions; bioelectric recordings

Special Issue Information

Dear Colleagues,

Mandelbrot proposed that some natural curves (as for instance the coastlines) can be modelled by means of a fractal methodology. This author studied the concept of self-similarity in depth, linking it with that of dimension, and finding hidden rules of many phenomena. He realized that some apparently erratic behaviors own an inherent organization that deserves to be discovered.

In this Special Issue, we wish to review different ways of defining self-similar curves, and study some of their properties. We want to revisit fundamental milestones of the origin and evolution of the fractal curves that, in some cases, agree with nowhere differentiable mappings but are not exhausted by them. Our main hypothesis is that many apparently random phenomena (climatic records, electrocardiograms, spread disease, etc.) can be successfully modelled by means of fractal functions. A vast bibliography confirms this assumption.

Prof. Dr. María Antonia Navascués
Prof. Dr. María Victoria Sebastián
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

Fractal interpolation functions
One-dimensional chaos
Fractal curves
Non-differentiability, fractional derivatives
Fractional Brownian motions
Fractal surfaces
Applications of fractal functions and fractal sets

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Published Papers (7 papers)

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Editorial

Jump to: Research

2 pages, 193 KiB

Open AccessEditorial

Special Issue: Fractal Functions and Applications

by María Antonia Navascués and María Victoria Sebastián

Fractal Fract. 2022, 6(8), 411; https://doi.org/10.3390/fractalfract6080411 - 26 Jul 2022

Viewed by 1193

Abstract

This volume gathers some important advances in the fields of fractional calculus and fractal curves and functions [...] Full article

(This article belongs to the Special Issue Fractal Functions and Applications)

Research

Jump to: Editorial

19 pages, 1036 KiB

Open AccessArticle

Multivariate Fractal Functions in Some Complete Function Spaces and Fractional Integral of Continuous Fractal Functions

by Kshitij Kumar Pandey and Puthan Veedu Viswanathan

Fractal Fract. 2021, 5(4), 185; https://doi.org/10.3390/fractalfract5040185 - 25 Oct 2021

Cited by 3 | Viewed by 2718

Abstract

There has been a considerable evolution of the theory of fractal interpolation function (FIF) over the last three decades. Recently, we introduced a multivariate analogue of a special class of FIFs, which is referred to as

α

-fractal functions, from the viewpoint of [...] Read more.

α

-fractal functions, from the viewpoint of approximation theory. In the current note, we continue our study on multivariate

α

-fractal functions, but in the context of a few complete function spaces. For a class of fractal functions defined on a hyperrectangle

Ω

in the Euclidean space

R^{n}

, we derive conditions on the defining parameters so that the fractal functions are elements of some standard function spaces such as the Lebesgue spaces

L^{p} (Ω)

, Sobolev spaces

W^{m, p} (Ω)

, and Hölder spaces

C^{m, σ} (Ω)

, which are Banach spaces. As a simple consequence, for some special choices of the parameters, we provide bounds for the Hausdorff dimension of the graph of the corresponding multivariate

α

-fractal function. We shall also hint at an associated notion of fractal operator that maps each multivariate function in one of these function spaces to its fractal counterpart. The latter part of this note establishes that the Riemann–Liouville fractional integral of a continuous multivariate

α

-fractal function is a fractal function of similar kind. Full article

(This article belongs to the Special Issue Fractal Functions and Applications)

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23 pages, 1018 KiB

Open AccessEditor’s ChoiceArticle

A Mathematical Study of a Coronavirus Model with the Caputo Fractional-Order Derivative

by Youcef Belgaid, Mohamed Helal, Abdelkader Lakmeche and Ezio Venturino

Fractal Fract. 2021, 5(3), 87; https://doi.org/10.3390/fractalfract5030087 - 3 Aug 2021

Cited by 11 | Viewed by 2603

Abstract

In this work, we introduce a minimal model for the current pandemic. It incorporates the basic compartments: exposed, and both symptomatic and asymptomatic infected. The dynamical system is formulated by means of fractional operators. The model equilibria are analyzed. The theoretical results indicate that their stability behavior is the same as for the corresponding system formulated via standard derivatives. This suggests that, at least in this case for the model presented here, the memory effects contained in the fractional operators apparently do not seem to play a relevant role. The numerical simulations instead reveal that the order of the fractional derivative has a definite influence on both the equilibrium population levels and the speed at which they are attained. Full article

(This article belongs to the Special Issue Fractal Functions and Applications)

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14 pages, 2733 KiB

Open AccessArticle

Iterated Functions Systems Composed of Generalized θ-Contractions

by Pasupathi Rajan, María A. Navascués and Arya Kumar Bedabrata Chand

Fractal Fract. 2021, 5(3), 69; https://doi.org/10.3390/fractalfract5030069 - 14 Jul 2021

Cited by 8 | Viewed by 2848

Abstract

The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of [...] Read more.

θ

-contraction IFS, which is a finite collection of generalized

θ

-contraction functions

T_{1}, \dots, T_{N}

from finite Cartesian product space

X \times \dots \times X

into X, where

(X, d)

is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued

θ

-contraction IFSs are Picard. Finally, when the map

θ

is continuous, we show the relation between the code space and the attractor of

θ

-contraction IFS. Full article

(This article belongs to the Special Issue Fractal Functions and Applications)

28 pages, 1616 KiB

Open AccessArticle

Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

by Gerd Baumann

Fractal Fract. 2021, 5(2), 43; https://doi.org/10.3390/fractalfract5020043 - 10 May 2021

Cited by 8 | Viewed by 4666

Abstract

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order. Full article

(This article belongs to the Special Issue Fractal Functions and Applications)

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15 pages, 447 KiB

Open AccessArticle

Fractal Frames of Functions on the Rectangle

by María A. Navascués, Ram Mohapatra and Md. Nasim Akhtar

Fractal Fract. 2021, 5(2), 42; https://doi.org/10.3390/fractalfract5020042 - 8 May 2021

Cited by 5 | Viewed by 2599

Abstract

In this paper, we define fractal bases and fractal frames of

L^{2} (I \times J)

, where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of [...] Read more.

In this paper, we define fractal bases and fractal frames of

L^{2} (I \times J)

, where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of

L^{2} (I \times J)

with the tensor product space

L^{2} (I) ⨂ L^{2} (J)

. First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of

L^{2} (I \times J)

composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable. Full article

(This article belongs to the Special Issue Fractal Functions and Applications)

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13 pages, 318 KiB

Open AccessArticle

Fractal Interpolation Using Harmonic Functions on the Koch Curve

by Song-Il Ri, Vasileios Drakopoulos and Song-Min Nam

Fractal Fract. 2021, 5(2), 28; https://doi.org/10.3390/fractalfract5020028 - 5 Apr 2021

Cited by 6 | Viewed by 2497

Abstract

The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions. Full article

(This article belongs to the Special Issue Fractal Functions and Applications)

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