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Axioms
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Fractal Analysis and Mathematical Integration
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Fractal Analysis and Mathematical Integration

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 30 September 2025 | Viewed by 2956

Special Issue Editors

Dr. Donatella Bongiorno

E-Mail Website
Guest Editor
Department of Engineering, University of Palermo, 90100 Palermo, Italy
Interests: fractal calculus; measure theory
Dr. Alireza Khalili Golmankhaneh

E-Mail Website
Guest Editor
Department of Physics, Urmia Branch, Islamic Azad University, Urmia 63896, West Azerbaijan, Iran
Interests: fractal calculus; fractal analysis

Special Issue Information

Dear Colleagues,

Fractal analysis is an emerging field that has significant applications across various domains, including physics, engineering and mathematics.

This Special Issue aims to comprehensively explore the latest developments in fractal analysis, in measure theory with particular attention given to the theory of integration and in fractal calculus that extends traditional calculus to fractal sets and curves with an algorithmic approach. The scope of this Special Issue is to show through the contributions of the best researchers in the field the developments in fractal calculus and real analysis with particular attention given to the theory of differentiation and integration.

Fractal calculus is a new branch of mathematics that, by an algorithmic approach, offers a generalization of traditional calculus. The environment of this calculation is the fractal subsets of the real line, like the classical Cantor set; the self-similar sets with overlap; the fractal subsets of the plane, like the Sierpinski gasket or the generalized Cantor-type sets; and finally the fractal curves, for instance, the von Koch curve.

This Special Issue will present both theoretical advancements and practical applications.

Dr. Donatella Bongiorno
Dr. Alireza Khalili Golmankhaneh
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. Axioms 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 2400 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 analysis
  • fractal calculus
  • measure theory
  • mathematical integration
  • fractal subsets

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (3 papers)

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Research

12 pages, 1032 KiB  
Article
Fractal Continuum Maxwell Creep Model
by Andriy Kryvko, Claudia del C. Gutiérrez-Torres, José Alfredo Jiménez-Bernal, Orlando Susarrey-Huerta, Eduardo Reyes de Luna and Didier Samayoa
Axioms 2025, 14(1), 33; https://doi.org/10.3390/axioms14010033 - 2 Jan 2025
Viewed by 698
Abstract
In this work, the fractal continuum Maxwell law for the creep phenomenon is introduced. By mapping standard integer space-time into fractal continuum space-time using the well-known Balankin’s approach to variable-order fractal calculus, the fractal version of Maxwell model is developed. This methodology employs [...] Read more.
In this work, the fractal continuum Maxwell law for the creep phenomenon is introduced. By mapping standard integer space-time into fractal continuum space-time using the well-known Balankin’s approach to variable-order fractal calculus, the fractal version of Maxwell model is developed. This methodology employs local fractional differential operators on discontinuous properties of fractal sets embedded in the integer space-time so that they behave as analytic envelopes of non-analytic functions in the fractal continuum space-time. Then, creep strain ε(t), creep modulus J(t), and relaxation compliance G(t) in materials with fractal linear viscoelasticity can be described by their generalized forms, εβ(t),Jβ(t) and Gβ(t), where β=dimS/dimH represents the time fractal dimension, and it implies the variable-order of fractality of the self-similar domain under study, which are dimS and dimH for their spectral and Hausdorff dimensions, respectively. The creep behavior depends on beta, which is characterized by its geometry and fractal topology: as beta approaches one, the fractal creep behavior approaches its standard behavior. To illustrate some physical implications of the suggested fractal Maxwell creep model, graphs that showcase the specific details and outcomes of our results are included in this study. Full article
(This article belongs to the Special Issue Fractal Analysis and Mathematical Integration)
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16 pages, 4018 KiB  
Article
Fractals as Julia Sets for a New Complex Function via a Viscosity Approximation Type Iterative Methods
by Ahmad Almutlg and Iqbal Ahmad
Axioms 2024, 13(12), 850; https://doi.org/10.3390/axioms13120850 - 3 Dec 2024
Cited by 2 | Viewed by 1002
Abstract
In this article, we examine and investigate various variants of Julia set patterns for complex exponential functions W(z)=αezn+βzm+logct, and [...] Read more.
In this article, we examine and investigate various variants of Julia set patterns for complex exponential functions W(z)=αezn+βzm+logct, and T(z)=αezn+βzm+γ (which are analytic except at z=0) where n≥2, m,n∈N, α,β,γ∈C,c∈C∖{0} and t∈R,t≥1, by employing a viscosity approximation-type iterative method. We employ the proposed iterative method to establish an escape criterion for visualizing Julia sets. We provide graphical illustrations of Julia sets that emphasize their sensitivity to different iteration parameters. We present graphical illustrations of Julia sets; the color, size, and shape of the images change with variations in the iteration parameters. With fixed input parameters, we observe the intriguing behavior of Julia sets for different values of n and m. We hope that the conclusions of this study will inspire researchers with an interest in fractal geometry. Full article
(This article belongs to the Special Issue Fractal Analysis and Mathematical Integration)
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20 pages, 459 KiB  
Article
Fractal Differential Equations of 2α-Order
by Alireza Khalili Golmankhaneh and Donatella Bongiorno
Axioms 2024, 13(11), 786; https://doi.org/10.3390/axioms13110786 - 14 Nov 2024
Viewed by 882
Abstract
In this research paper, we provide a concise overview of fractal calculus applied to fractal sets. We introduce and solve a 2α-order fractal differential equation with constant coefficients across different scenarios. We propose a uniqueness theorem for 2α-order fractal [...] Read more.
In this research paper, we provide a concise overview of fractal calculus applied to fractal sets. We introduce and solve a 2α-order fractal differential equation with constant coefficients across different scenarios. We propose a uniqueness theorem for 2α-order fractal linear differential equations. We define the solution space as a vector space with non-integer orders. We establish precise conditions for 2α-order fractal linear differential equations and derive the corresponding fractal adjoint differential equation. Full article
(This article belongs to the Special Issue Fractal Analysis and Mathematical Integration)
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