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Axioms
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Fractional Differential Equation and Its Applications
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Fractional Differential Equation and Its Applications

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: 31 December 2025 | Viewed by 4283

Special Issue Editors

Dr. Jiankang Liu

E-Mail Website
Guest Editor
School of applied science, Taiyuan University of Science and Technology, Taiyuan, China
Interests: fractional differential equations; stochastic differential equations; impulsive differential equations; stochastic dynamics; stochastic analysis; nonlinear dynamics
Dr. Shuo Zhang

E-Mail Website
Guest Editor
School of mathematics and statistics, Northwestern Polytechnical University, Xi'an 710072, China
Interests: applied mathematics; nonlinear dynamics; control; information science; neural network; complex network system
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In recent years, the research on fractional differential equations and their applications has flourished. In particular, there have been a large number of outstanding achievements in approximation theory, dynamical behavior, efficient numerical algorithms, complex networks, parameter identification, and other aspects of fractional differential systems, showing a good development trend. Based on this, this Special Issue is dedicated to collecting the latest excellent works on the theory, methods, and applications of fractional differential equations.

In this Special Issue, original research articles and reviews are welcome. Research areas may include (but are not limited to) the following:

  • Fractional differential equations;
  • Stochastic fractional differential equations;
  • Fractional delay differential equations;
  • Fractional complex networks;
  • Fractional hybrid differential equations;
  • Differential equations driven by fractional Brownian motion;
  • Identification methods for fractional differential systems;
  • Modeling, method, and analysis of fractional differential systems;
  • Nonlinear dynamics of fractional differential equations;
  • We look forward to receiving your contributions;

Dr. Jiankang Liu
Dr. Shuo 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. 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

  • fractional differential equations
  • stability analysis
  • bifurcation, chaos, and synchronization
  • identification
  • modeling
  • well-posedness
  • fractional Brownian motion
  • noise
  • networks
  • delay, impulse, and perturbance

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

Published Papers (5 papers)

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Research

20 pages, 5488 KiB  
Article
Circuit Design and Implementation of a Time-Varying Fractional-Order Chaotic System
by Burak Arıcıoğlu
Axioms 2025, 14(4), 310; https://doi.org/10.3390/axioms14040310 - 17 Apr 2025
Viewed by 208
Abstract
This paper presents a circuit design methodology for the analog realization of time-varying fractional-order chaotic systems. While most existing studies implement such systems by switching between two or more constant fractional orders, these approaches become impractical when the fractional order changes smoothly over [...] Read more.
This paper presents a circuit design methodology for the analog realization of time-varying fractional-order chaotic systems. While most existing studies implement such systems by switching between two or more constant fractional orders, these approaches become impractical when the fractional order changes smoothly over time. To overcome this limitation, the proposed method introduces a transfer function approximation specifically designed for variable fractional-order integrators. The formulation relies on a linear and time-invariant definition of the fractional-order operator, ensuring compatibility with Laplace-domain analysis. Under the condition that the fractional-order function is Laplace-transformable and its Bode plot slope lies between −20 dB/decade and 0 dB/decade, the system is realized using op-amps and standard RC components. The Grünwald–Letnikov method is employed for numerical calculation of phase portraits, which are then compared with simulation and experimental results. The strong agreement among these results confirms the effectiveness of the proposed method. Full article
(This article belongs to the Special Issue Fractional Differential Equation and Its Applications)
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17 pages, 276 KiB  
Article
Blow-Up of Solutions in a Fractionally Damped Plate Equation with Infinite Memory and Logarithmic Nonlinearity
by Muhammad Fahim Aslam, Jianghao Hao, Salah Boulaaras and Luqman Bashir
Axioms 2025, 14(2), 80; https://doi.org/10.3390/axioms14020080 - 22 Jan 2025
Viewed by 683
Abstract
In this article, we consider the dynamics of a viscoelastic plate equation with internal fractional damping, a nonlinear logarithmic source, and infinite memory effects. The existence of a local weak solution is shown effectively through the framework of semigroup theory. Furthermore, we show [...] Read more.
In this article, we consider the dynamics of a viscoelastic plate equation with internal fractional damping, a nonlinear logarithmic source, and infinite memory effects. The existence of a local weak solution is shown effectively through the framework of semigroup theory. Furthermore, we show that the blow-up in finite time of the local solution may occur under specific conditions and is demonstrated within the development of a suitable Lyapunov functional. Our result offers an insight into the challenges presented by this class of equations and their relevance to physical systems. Full article
(This article belongs to the Special Issue Fractional Differential Equation and Its Applications)
15 pages, 2338 KiB  
Article
A Comparative Study and Numerical Solutions for the Fractional Modified Lorenz–Stenflo System Using Two Methods
by Mohamed Elbadri, Mohamed A. Abdoon, Abdulrahman B. M. Alzahrani, Rania Saadeh and Mohammed Berir
Axioms 2025, 14(1), 20; https://doi.org/10.3390/axioms14010020 - 30 Dec 2024
Cited by 1 | Viewed by 607
Abstract
This paper provides a solution to the new fractional-order Lorenz–Stenflo model using the adaptive predictor–corrector approach and the ρ-Laplace New Iterative Method (LρNIM), representing an extensive comparison between both techniques with RK4 related to accuracy and [...] Read more.
This paper provides a solution to the new fractional-order Lorenz–Stenflo model using the adaptive predictor–corrector approach and the ρ-Laplace New Iterative Method (LρNIM), representing an extensive comparison between both techniques with RK4 related to accuracy and error analysis. The results show that the suggested approaches allow one to be more accurate in analyzing the dynamics of the system. These techniques also produce results that are comparable to the results of other approximate techniques. The techniques can, thus, be used on a wider class of systems in order to provide more accurate results. These techniques also appropriately identify chaotic attractors in the system. These techniques can be applied to solve various numerical problems arising in science and engineering in the future. Full article
(This article belongs to the Special Issue Fractional Differential Equation and Its Applications)
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16 pages, 7742 KiB  
Article
Response Analysis and Vibration Suppression of Fractional Viscoelastic Shape Memory Alloy Spring Oscillator Under Harmonic Excitation
by Rong Guo, Na Meng, Jinling Wang, Junlin Li and Jinbin Wang
Axioms 2024, 13(11), 803; https://doi.org/10.3390/axioms13110803 - 19 Nov 2024
Viewed by 696
Abstract
This study investigates the dynamic behavior and vibration mitigation of a fractional single-degree-of-freedom (SDOF) viscoelastic shape memory alloy spring oscillator system subjected to harmonic external forces. A fractional derivative approach is employed to characterize the viscoelastic properties of shape memory alloy materials, leading [...] Read more.
This study investigates the dynamic behavior and vibration mitigation of a fractional single-degree-of-freedom (SDOF) viscoelastic shape memory alloy spring oscillator system subjected to harmonic external forces. A fractional derivative approach is employed to characterize the viscoelastic properties of shape memory alloy materials, leading to the development of a novel fractional viscoelastic model. The model is then theoretically examined using the averaging method, with its effectiveness being confirmed through numerical simulations. Furthermore, the impact of various parameters on the system’s low- and high-amplitude vibrations is explored through a visual response analysis. These findings offer valuable insights for applying fractional sliding mode control (SMC) theory to address the system’s vibration control challenges. Despite the high-amplitude vibrations induced by the fractional order, SMC effectively suppresses these vibrations in the shape memory alloy spring system, thereby minimizing the risk of catastrophic events. Full article
(This article belongs to the Special Issue Fractional Differential Equation and Its Applications)
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25 pages, 444 KiB  
Article
Ulam–Hyers Stability and Simulation of a Delayed Fractional Differential Equation with Riemann–Stieltjes Integral Boundary Conditions and Fractional Impulses
by Xiaojun Lv, Kaihong Zhao and Haiping Xie
Axioms 2024, 13(10), 682; https://doi.org/10.3390/axioms13100682 - 1 Oct 2024
Cited by 1 | Viewed by 983
Abstract
In this article, we delve into delayed fractional differential equations with Riemann–Stieltjes integral boundary conditions and fractional impulses. By using differential inequality techniques and some fixed-point theorems, some novel sufficient assessments for convenient verification have been devised to ensure the existence and uniqueness [...] Read more.
In this article, we delve into delayed fractional differential equations with Riemann–Stieltjes integral boundary conditions and fractional impulses. By using differential inequality techniques and some fixed-point theorems, some novel sufficient assessments for convenient verification have been devised to ensure the existence and uniqueness of solutions. We further employ the nonlinear analysis to reveal that this problem is Ulam–Hyers (UH) stable. Finally, some examples and numerical simulations are presented to illustrate the reliability and validity of our main results. Full article
(This article belongs to the Special Issue Fractional Differential Equation and Its Applications)
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