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Axioms
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Advances in Fractional-Order Difference and Differential Equations
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Advances in Fractional-Order Difference and Differential Equations

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

Deadline for manuscript submissions: 20 July 2026 | Viewed by 1796

Special Issue Editors

Dr. Azhar Ali Zafar

E-Mail Website
Guest Editor
Department of Mathematics, Government College University, Lahore 54770, Pakistan
Interests: research in dynamical systems; vibrating systems; chaos and bifurcation; guitar playing
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Nehad Ali Shah

E-Mail Website
Guest Editor
Department of Mechanical Engineering, Sejong University, Seoul 05006, Republic of Korea
Interests: fluid dynamics (Newtonian and non-Newtonian fluids; heat and mass transfer; viscoelastic models with memory; fractional thermoelasticity)
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional-order difference and differential equations serve as robust tools for modeling, analyzing, and controlling systems characterized by memory effects, non-local interactions, and complex dynamic behavior, making them essential in contemporary applied mathematics and engineering. This Special Issue, “Advances in Fractional-Order Difference and Differential Equations”, functions as a focused knowledge platform, facilitating the development and application of advanced methodologies while fostering collaboration and knowledge exchange within the global research community.

Dr. Azhar Ali Zafar
Prof. Dr. Nehad Ali Shah
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 250 words) can be sent to the Editorial Office for assessment.

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 calculus
  • non-integer order systems
  • memory effects
  • non-local operators
  • anomalous diffusion
  • hereditary systems
  • control systems
  • signal processing
  • viscoelastic materials
  • biological systems
  • heat and mass transfer
  • financial modeling
  • image processing
  • epidemiological modeling
  • discrete dynamical systems
  • operator theory
  • stability analysis
  • existence and uniqueness
  • Laplace transform method
  • Adomian decomposition
  • finite difference schemes
  • predictor–corrector method

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Published Papers (3 papers)

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26 pages, 2199 KB  
Article
Analytical and Numerical Analysis of Multidimensional Diffusion Processes in L2 Space Under Third-Kind Boundary Conditions
by Zafar Duman Abbasov, Ghadah Albeladi, Mohamed Gamal and Youssri Hassan Youssri
Axioms 2026, 15(5), 380; https://doi.org/10.3390/axioms15050380 - 19 May 2026
Viewed by 145
Abstract
This research paper investigates the solution of diffusion equations characterized by Third-Kind (Robin) boundary conditions within n-dimensional complex domains. The analysis is conducted in the L2 Hilbert space, which facilitates the substantiation of both the existence and uniqueness of solutions through [...] Read more.
This research paper investigates the solution of diffusion equations characterized by Third-Kind (Robin) boundary conditions within n-dimensional complex domains. The analysis is conducted in the L2 Hilbert space, which facilitates the substantiation of both the existence and uniqueness of solutions through variational methods. Analytical solutions are derived for multidimensional domains by employing the Fourier method and spectral analysis techniques. Complementing this theoretical framework, a high-accuracy numerical approach based on the Associated Legendre Polynomials Collocation Spectral Method (ALP-CSM) with Chebyshev–Gauss–Lobatto nodes is developed. Rigorous convergence analysis confirms spectral accuracy, with numerical examples in one, two, and three dimensions demonstrating error decay from O(10−3) to machine precision O(10−15). The mathematical impact of Third-Kind boundary conditions on the diffusion rate and the steady state of the system is demonstrated. The obtained results provide a robust tool for modeling physical processes, particularly in systems involving heat exchange on the surfaces of complex-structured domains, offering both theoretical insight and computational efficiency. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Difference and Differential Equations)
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16 pages, 317 KB  
Article
Solution Operators for Caputo-Type Fractional Evolution Equations with Damping
by Ting-Ting Hu, Shi-You Lin and Zhi-Chao Lu
Axioms 2026, 15(3), 179; https://doi.org/10.3390/axioms15030179 - 28 Feb 2026
Viewed by 479
Abstract
This paper investigates an abstract fractional Cauchy problem with damping formulated in the sense of the Caputo derivative, where the derivative orders satisfy 0<δ<γ≤1. By introducing the concept of a Caputo fractional [...] Read more.
This paper investigates an abstract fractional Cauchy problem with damping formulated in the sense of the Caputo derivative, where the derivative orders satisfy 0<δ<γ≤1. By introducing the concept of a Caputo fractional (γ,δ,k) resolvent and systematically analyzing its fundamental properties, together with key features of the generalized Mittag–Leffler (ML) function, we establish the uniqueness and existence of strong solutions for this class of damped fractional-order evolution equations. Under more restrictive assumptions on the underlying operators, the solution admits an explicit representation in terms of ML-type functions associated with fractional exponents. Furthermore, we demonstrate that the proposed abstract framework can be effectively applied to concrete models, including fractional diffusion equations with damping. These results highlight the relevance and necessity of fractional damping models in accurately describing complex dynamical phenomena, such as vibration processes and anomalous diffusion. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Difference and Differential Equations)
25 pages, 2019 KB  
Article
Statistical Convergence for Grünwald–Letnikov Fractional Differences: Stability, Approximation, and Diagnostics in Fuzzy Normed Spaces
by Hasan Öğünmez and Muhammed Recai Türkmen
Axioms 2025, 14(10), 725; https://doi.org/10.3390/axioms14100725 - 25 Sep 2025
Cited by 2 | Viewed by 676
Abstract
We present a unified framework for fuzzy statistical convergence of Grünwald–Letnikov (GL) fractional differences in Bag–Samanta fuzzy normed linear spaces, addressing memory effects and nonlocality inherent to fractional-order models. Theoretically, we establish the uniqueness, linearity, and invariance of fuzzy statistical limits and prove [...] Read more.
We present a unified framework for fuzzy statistical convergence of Grünwald–Letnikov (GL) fractional differences in Bag–Samanta fuzzy normed linear spaces, addressing memory effects and nonlocality inherent to fractional-order models. Theoretically, we establish the uniqueness, linearity, and invariance of fuzzy statistical limits and prove a Cauchy characterization: fuzzy statistical convergence implies fuzzy statistical Cauchyness, while the converse holds in fuzzy-complete spaces (and in the completion, otherwise). We further develop an inclusion theory linking fuzzy strong Cesàro summability—including weighted means—to fuzzy statistical convergence. Via the discrete Q-operator, all statements transfer verbatim between nabla-left and delta-right GL forms, clarifying the binomial GL↔discrete Riemann–Liouville correspondence. Beyond structure, we propose density-based residual diagnostics for GL discretizations of fractional initial-value problems: when GL residuals are fuzzy statistically negligible, trajectories exhibit Ulam–Hyers-type robustness in the fuzzy topology. We also formulate a fuzzy Korovkin-type approximation principle under GL smoothing: Cesàro control on the test set {1,x,x2} propagates to arbitrary targets, yielding fuzzy statistical convergence for positive-operator sequences. Worked examples and an engineering-style case study (thermal balance with memory and bursty disturbances) illustrate how the diagnostics certify robustness of GL numerical schemes under sparse spikes and imprecise data. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Difference and Differential Equations)
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