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Fractal Fract
Special Issues
Fractal Functions: Theoretical Research and Application Analysis
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Fractal Functions: Theoretical Research and Application Analysis

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

Deadline for manuscript submissions: 31 October 2025 | Viewed by 2790

Special Issue Editors

Prof. Dr. Yong-Shun Liang

E-Mail Website
Guest Editor
School of Mathematics and Statistics; Nanjing University of Science and Technology, Nanjing 210094, China
Interests: fractional calculus on fractal functions; fractal geometry; fractal dimensions; function approximation theory
Prof. Dr. Jia Liu

E-Mail Website
Guest Editor
Institute of statistics and applied mathematics, Anhui University of Finance and Economics, Bengbu 233030, China
Interests: fractal geometry; fractal functions; number theory

Special Issue Information

Dear Colleagues,

Fractal geometry is an important branch of mathematics that allows for the description of sets that are too intricate to fit into classical geometry. The concept of fractals was first introduced by Mandelbrot in the 1970s as a class of highly irregular sets, usually presenting with self-similarity, infinite complexity, and a non-integral fractal dimension. Up to now, it has been hugely significant for the development of a variety of sciences. In mathematics, fractals originate from chaos and dynamic systems. Soon after their discovery, they began to appear in almost every field and were systematically studied using classical and modern methods. In recent years, scholars have mainly focused on the following research objects: fractal dimensions of graphs, fractal interpolation and approximation, fractals and dynamical systems, self-similarity and Lipschitz equivalence, geometric measure theory, fractional calculus of fractal functions, fractal geometry, number theory, etc.

Fractal curves and fractal functions are a class of important research objects in fractal geometry. As is well known, fractal curves are widely distributed in nature, being found in lightning, snowflakes, coastlines, geological crack lines, and so on. In mathematical analysis, fractal functions are usually regarded as continuous functions of one variable based on a two-dimensional Cartesian coordinate system to probe their fractal characteristics. The fractal dimension, as a common measure of the geometric complexity of sets, can be an essential tool to describe their fractal characteristics. In recent years, fractal functions have been widely applied in other academic fields, such as physics, statistics, geology, material science, quantization theory, signal processing, computer image processing, pattern recognition, and more. Therefore, fractal functions have increasingly shown their tremendous research value for both real life and scientific development.

This Special Issue aims to collect a series of high-quality papers from renowned experts around the world to present the latest research on fractal functions with their application and analysis in various fields.

Prof. Dr. Yong-Shun Liang
Prof. Dr. Jia Liu
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

  • iterated function systems
  • fractal interpolation functions
  • fractal functions
  • fractal surfaces
  • fractional calculus
  • function spaces
  • self-similar sets and measures
  • self-affine and self-conformal sets/measures
  • fractal approximation and modeling
  • quantization dimension
  • box dimension, Hausdorff dimension, and L^q dimensions
  • applications of the fractal functions to any areas of finance, engineering, economy, biology, etc.

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

Published Papers (4 papers)

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Research

15 pages, 516 KiB  
Article
Remarks on the Relationship Between Fractal Dimensions and Convergence Speed
by Jiaqi Qiu and Yongshun Liang
Fractal Fract. 2025, 9(5), 303; https://doi.org/10.3390/fractalfract9050303 - 6 May 2025
Viewed by 471
Abstract
This paper conducts an in-depth investigation into the fundamental relationship between the fractal dimensions and convergence properties of mathematical sequences. By concentrating on three representative classes of sequences, namely, the factorial-decay, logarithmic-decay, and factorial–exponential types, a comprehensive framework is established to link their [...] Read more.
This paper conducts an in-depth investigation into the fundamental relationship between the fractal dimensions and convergence properties of mathematical sequences. By concentrating on three representative classes of sequences, namely, the factorial-decay, logarithmic-decay, and factorial–exponential types, a comprehensive framework is established to link their geometric characteristics with asymptotic behavior. This study makes two significant contributions to the field of fractal analysis. Firstly, a unified methodology is developed for the calculation of multiple fractal dimensions, including the Box, Hausdorff, Packing, and Assouad dimensions, of discrete sequences. This methodology reveals how these dimensional quantities jointly describe the structures of sequences, providing a more comprehensive understanding of their geometric properties. Secondly, it is demonstrated that different fractal dimensions play distinct yet complementary roles in regulating convergence rates. Specifically, the Box dimension determines the global convergence properties of sequences, while the Assouad dimension characterizes the local constraints on the speed of convergence. The theoretical results presented herein offer novel insights into the inherent connection between geometric complexity and analytical behavior within sequence spaces. These findings have immediate and far-reaching implications for various applications that demand precise control over convergence properties, such as numerical algorithm design and signal processing. Notably, the identification of dimension-based convergence criteria provides practical and effective tools for the analysis of sequence behavior in both pure mathematical research and applied fields. Full article
(This article belongs to the Special Issue Fractal Functions: Theoretical Research and Application Analysis)
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15 pages, 560 KiB  
Article
Entropiesand Degree-Based Topological Indices of Generalized Sierpiński Graphs
by Si-Ao Xu, Jia-Dong Si and Jia-Bao Liu
Fractal Fract. 2025, 9(3), 190; https://doi.org/10.3390/fractalfract9030190 - 19 Mar 2025
Viewed by 453
Abstract
Fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shannon entropy) that is used [...] Read more.
Fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shannon entropy) that is used to describe the topological structural complexity or degree of disorder in networks. Topological indices, as graph invariants, provide quantitative descriptors for characterizing global structural properties. In this paper, we investigate two types of generalized Sierpiński graphs constructed on the basis of different seed graphs, and employ six topological indices—the first Zagreb index, the second Zagreb index, the forgotten index, the augmented Zagreb index, the Sombor index, and the elliptic Sombor index—to analyze the corresponding entropy. We utilize the method of edge partition based on vertex degrees and derive analytical formulations for the first Zagreb entropy, the second Zagreb entropy, the forgotten entropy, the augmented Zagreb entropy, the Sombor entropy, and the elliptic Sombor entropy. This research approach, which integrates entropy with Sierpiński network characteristics, furnishes novel perspectives and instrumental tools for addressing challenges in chemical graph theory, computer networks, and other related fields. Full article
(This article belongs to the Special Issue Fractal Functions: Theoretical Research and Application Analysis)
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13 pages, 262 KiB  
Article
Progress on Fractal Dimensions of the Weierstrass Function and Weierstrass-Type Functions
by Yue Qiu and Yongshun Liang
Fractal Fract. 2025, 9(3), 143; https://doi.org/10.3390/fractalfract9030143 - 25 Feb 2025
Viewed by 905
Abstract
The Weierstrass function W(x)=∑n=1∞ancos(2πbnx) is a function that is continuous everywhere and differentiable nowhere. There are many investigations on fractal dimensions of the Weierstrass [...] Read more.
The Weierstrass function W(x)=∑n=1∞ancos(2πbnx) is a function that is continuous everywhere and differentiable nowhere. There are many investigations on fractal dimensions of the Weierstrass function, and the investigation of its Hausdorff dimension is still ongoing. In this paper, we summarize past researchers’ investigations on fractal dimensions of the Weierstrass function graph. Full article
(This article belongs to the Special Issue Fractal Functions: Theoretical Research and Application Analysis)
20 pages, 1031 KiB  
Article
Entropies and Degree-Based Topological Indices of Coronene Fractal Structures
by Si-Ao Xu and Jia-Bao Liu
Fractal Fract. 2025, 9(3), 133; https://doi.org/10.3390/fractalfract9030133 - 20 Feb 2025
Cited by 1 | Viewed by 514
Abstract
Molecular fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shannon entropy) and is [...] Read more.
Molecular fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shannon entropy) and is used to describe the topological structural complexity or degree of disorder in networks. A topological index is a numeric quantity associated with a network or a graph that characterizes its whole structural properties. In this study, we focus on fractal structures formed by systematically repeating a fixed unit of coronene, a polycyclic aromatic hydrocarbon composed of six benzene rings fused in a hexagonal pattern. In this paper, three types of coronal fractal structures, namely zigzag (ZHCF), armchair (AHCF), and rectangular (RCF), are studied, and their five degree-based topological indices and corresponding entropies are calculated. Full article
(This article belongs to the Special Issue Fractal Functions: Theoretical Research and Application Analysis)
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