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Fractal Fract
Special Issues
Variable-Order Fractional Problems: Modeling, Analysis, Approximation and Application
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Variable-Order Fractional Problems: Modeling, Analysis, Approximation and Application

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 (30 November 2022) | Viewed by 10336

Special Issue Editors

Dr. Xiangcheng Zheng

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Guest Editor
School of Mathematical Sciences, Peking University, Beijing 100871, China
Interests: fractional calculus; fractional differential equation; variable-order; numerical method; mathematical analysis
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. HongGuang Sun

E-Mail
Guest Editor
1. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, Jiangsu, China
2. Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 210098, Jiangsu, China
Interests: variable-order fractional derivative; anomalous diffusion; power law; solute transport; meshless method; analytical solution
Prof. Dr. Hong Wang

E-Mail Website
Guest Editor
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
Interests: fractional differential equation; analysis; numerical discretization
Prof. Dr. Yong Zhang

E-Mail Website
Guest Editor
Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USA
Interests: stochastic hydrology; contaminant transport; fractional-derivative model development and application

Special Issue Information

Dear Colleagues,

Variable-order fractional problems have attracted increasing attention in recent decades, with growing successful applications in various fields. Compared with their constant-order fractional analogues, the variability of the fractional order provides an extra dimension to improve the modeling capability of these models for complex phenomena. Furthermore, one could connect the fractional problems and their integer-order counterparts by adjusting the variable fractional order. However, the introduction of the variable order in fractional models leads to several mathematical and numerical difficulties that have not been previously encountered, and corresponding studies are far from well-developed.

This Special Issue aims to promote the investigation of variable-order fractional problems from all aspects, such as modeling, numerical methods and analysis, theoretical analysis, and applications. We invite you to submit comprehensive review papers and original articles. This issue will cover topics of interest including, but not limited to, the following topics:

  1. Modeling by equations involving variable-order fractional operators;
  2. Numerical discretization and numerical analysis for variable-order fractional problems;
  3. Mathematical analysis for variable-order fractional problems, e.g., well-posedness and smoothing properties of the solutions.
  4. Practical applications of variable-order fractional problems in all fields.
  5. Other related topics on variable-order fractional problems, e.g., optimal control problems, inverse problems, and calculus of variations.

Dr. Xiangcheng Zheng
Prof. Hongguang Sun
Prof. Hong Wang
Prof. Yong Zhang
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

  • variable-order
  • fractional calculus
  • fractional differential equation
  • modeling and application
  • approximation method
  • mathematical analysis
  • numerical analysis
  • numerical simulation

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

Published Papers (5 papers)

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Research

11 pages, 286 KiB  
Article
Analysis of a Hidden-Memory Variably Distributed-Order Time-Fractional Diffusion Equation
by Jinhong Jia
Fractal Fract. 2022, 6(11), 627; https://doi.org/10.3390/fractalfract6110627 - 28 Oct 2022
Viewed by 1208
Abstract
We analyze the well-posedness and regularity of a variably distributed-order time-fractional diffusion equation (tFDE) with a hidden-memory fractional derivative, which provide a competitive means to describe the anomalously diffusive transport of particles in heterogeneous media. We prove that the solution of a variably [...] Read more.
We analyze the well-posedness and regularity of a variably distributed-order time-fractional diffusion equation (tFDE) with a hidden-memory fractional derivative, which provide a competitive means to describe the anomalously diffusive transport of particles in heterogeneous media. We prove that the solution of a variably distributed-order tFDE has weak singularity at the initial time t=0 which depends on the upper bound of a distributed order α¯(0). Full article
(This article belongs to the Special Issue Variable-Order Fractional Problems: Modeling, Analysis, Approximation and Application)
25 pages, 8190 KiB  
Article
Circuit Implementation of Variable-Order Scaling Fractal-Ladder Fractor with High Resolution
by Bo Yu, Yifei Pu, Qiuyan He and Xiao Yuan
Fractal Fract. 2022, 6(7), 388; https://doi.org/10.3390/fractalfract6070388 - 12 Jul 2022
Cited by 3 | Viewed by 1852
Abstract
Extensive research has been conducted on the scaling fractal fractor using various structures. The development of high-resolution emulator circuits to achieve a variable-order scaling fractal fractor with high resolution is a major area of interest. We present a scaling fractal-ladder circuit for achieving [...] Read more.
Extensive research has been conducted on the scaling fractal fractor using various structures. The development of high-resolution emulator circuits to achieve a variable-order scaling fractal fractor with high resolution is a major area of interest. We present a scaling fractal-ladder circuit for achieving high-resolution variable-order fractor based on scaling expansion theory using a high-resolution multiplying digital-to-analog converter (HMDAC). Firstly, the circuit configuration of variable-order scaling fractal-ladder fractor (VSFF) is designed. A theoretical demonstration proves that VSFF exhibits the operational characteristics of variable-order fractional calculus. Secondly, a programmable resistor–capacitor series circuit and universal electronic component emulators are developed based on the HMDAC to adjust the resistance and capacitance in the circuit configuration. Lastly, the model, component parameters, approximation performance, and variable-order characteristics are analyzed, and the circuit is physically implemented. The experimental results demonstrate that the circuit exhibits variable-order characteristics, with an operational order ranging from −0.7 to −0.3 and an operational frequency ranging from 7.72Hz to 4.82kHz. The peak value of the input signal is 10V. This study also proposes a novel method for variable-order fractional calculus based on circuit theory. This study was the first attempt to implement feasible high-resolution continuous variable-order fractional calculus hardware based on VSFF. Full article
(This article belongs to the Special Issue Variable-Order Fractional Problems: Modeling, Analysis, Approximation and Application)
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21 pages, 376 KiB  
Article
The Traveling Wave Solutions in a Mixed-Diffusion Epidemic Model
by Ru Hou and Wen-Bing Xu
Fractal Fract. 2022, 6(4), 217; https://doi.org/10.3390/fractalfract6040217 - 11 Apr 2022
Cited by 2 | Viewed by 1762
Abstract
In this paper, we study the traveling wave solution of an epidemic model with mixed diffusion. First, we give two definitions of the minimum wave speeds and prove that they are equivalent. Second, the existence, decaying behavior, and uniqueness of traveling wave fronts [...] Read more.
In this paper, we study the traveling wave solution of an epidemic model with mixed diffusion. First, we give two definitions of the minimum wave speeds and prove that they are equivalent. Second, the existence, decaying behavior, and uniqueness of traveling wave fronts are obtained. Third, the signs of minimum wave speeds are studied, and further, in two specific cases of the dispersal kernel, we show how to identify the signs of minimum wave speeds. Full article
(This article belongs to the Special Issue Variable-Order Fractional Problems: Modeling, Analysis, Approximation and Application)
16 pages, 400 KiB  
Article
On Variable-Order Fractional Discrete Neural Networks: Solvability and Stability
by Amel Hioual, Adel Ouannas, Taki-Eddine Oussaeif, Giuseppe Grassi, Iqbal M. Batiha and Shaher Momani
Fractal Fract. 2022, 6(2), 119; https://doi.org/10.3390/fractalfract6020119 - 18 Feb 2022
Cited by 30 | Viewed by 2332
Abstract
Few papers have been published to date regarding the stability of neural networks described by fractional difference operators. This paper makes a contribution to the topic by presenting a variable-order fractional discrete neural network model and by proving its Ulam–Hyers stability. In particular, [...] Read more.
Few papers have been published to date regarding the stability of neural networks described by fractional difference operators. This paper makes a contribution to the topic by presenting a variable-order fractional discrete neural network model and by proving its Ulam–Hyers stability. In particular, two novel theorems are illustrated, one regarding the existence of the solution for the proposed variable-order network and the other regarding its Ulam–Hyers stability. Finally, numerical simulations of three-dimensional and two-dimensional variable-order fractional neural networks were carried out to highlight the effectiveness of the conceived theoretical approach. Full article
(This article belongs to the Special Issue Variable-Order Fractional Problems: Modeling, Analysis, Approximation and Application)
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7 pages, 252 KiB  
Article
Analysis of a Time-Fractional Substantial Diffusion Equation of Variable Order
by Xiangcheng Zheng, Hong Wang and Xu Guo
Fractal Fract. 2022, 6(2), 114; https://doi.org/10.3390/fractalfract6020114 - 15 Feb 2022
Cited by 1 | Viewed by 1821
Abstract
A time-fractional substantial diffusion equation of variable order is investigated, in which the variable-order fractional substantial derivative accommodates the memory effects and the structure change of the surroundings of the physical processes with respect to time. The existence and uniqueness of the solutions [...] Read more.
A time-fractional substantial diffusion equation of variable order is investigated, in which the variable-order fractional substantial derivative accommodates the memory effects and the structure change of the surroundings of the physical processes with respect to time. The existence and uniqueness of the solutions to the proposed model are proved, based on which the weighted high-order regularity of the solutions, in which the weight function characterizes the singularity of the solutions, are analyzed. Full article
(This article belongs to the Special Issue Variable-Order Fractional Problems: Modeling, Analysis, Approximation and Application)
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