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Mathematics
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Recent Advances in Fractal and Fractional Calculus
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Recent Advances in Fractal and Fractional Calculus

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: 30 November 2026 | Viewed by 2842

Editor

Prof. Dr. Marko Kostić

E-Mail Website
Guest Editor
Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, Serbia
Interests: functional analysis; fractional calculus; fractional integro-differential equations; fractional integro-difference equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The theories of fractal geometry, fractional calculus and fractional differential equations are active fields of research for many mathematicians. Fractional calculus and fractional differential equations, which emerged as the most important field of applied mathematics in the recent century, can be viewed as a special part of the theory of (abstract) Volterra integro-differential equations.

Additionally, discrete fractional calculus, discrete fractional differential equations and discrete Volterra equations are rapidly growing fields of research. Discrete fractional calculus is important in the modeling of various phenomena concerning complex dynamic systems, frequency response analysis, image processing, interval-valued systems and neural networks, etc.

The main purpose of this Special Issue is to present the recent developments in the theory of fractals, fractional calculus and fractional integro-differential and integro-difference equations. Thus, we encourage authors to submit papers on the applications of fractals and fractional integro-differential equations.

Prof. Dr. Marko Kostić
Guest Editor

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-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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
  • fractional differential equations
  • fractional difference equations
  • multidimensional fractional calculus
  • fractal geometry

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Published Papers (3 papers)

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Research

16 pages, 1341 KB  
Article
h-Stability of Nonlinear Hilfer Nabla Fractional Difference Equations
by Marko Kostić, Halis Can Koyuncuoğlu and Jagan Mohan Jonnalagadda
Mathematics 2026, 14(12), 2101; https://doi.org/10.3390/math14122101 - 11 Jun 2026
Viewed by 175
Abstract
This paper investigates the h-stability of nonlinear Hilfer nabla fractional difference equations, which interpolate between the Riemann–Liouville and Caputo nabla fractional differences through an additional type parameter. To the best of our knowledge, this is the first work devoted to the h-stability analysis [...] Read more.
This paper investigates the h-stability of nonlinear Hilfer nabla fractional difference equations, which interpolate between the Riemann–Liouville and Caputo nabla fractional differences through an additional type parameter. To the best of our knowledge, this is the first work devoted to the h-stability analysis of nonlinear Hilfer nabla fractional difference equations. By employing the properties of generalized nabla fractional sums and Taylor monomials together with a comparison-based approach, we establish new sufficient conditions guaranteeing the h-stability of the zero solution under suitable growth assumptions on the nonlinear term. Furthermore, we show that the associated solution map is differentiable with respect to the initial condition, providing a sensitivity framework for Hilfer-type discrete fractional systems. As another contribution, we derive a discrete variation in parameters formula for perturbed Hilfer nabla fractional difference equations, yielding an explicit representation of perturbed solutions. The obtained results extend the existing stability theory for discrete fractional systems and provide a unified framework encompassing both the Riemann–Liouville and Caputo nabla settings as particular cases. Full article
(This article belongs to the Special Issue Recent Advances in Fractal and Fractional Calculus)
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43 pages, 11364 KB  
Article
Mathematical Modeling of Neural Dynamics Through Stochastic Fractional FitzHugh–Nagumo Equations: An Inverse Problem Approach
by Dilara Altan Koç
Mathematics 2026, 14(5), 795; https://doi.org/10.3390/math14050795 - 26 Feb 2026
Viewed by 966
Abstract
Neural field dynamics in the cerebral cortex exhibit complex spatiotemporal patterns inadequately captured by classical integer-order diffusion models that assume exponentially decaying spatial interactions. This study establishes a stochastic fractional FitzHugh–Nagumo framework incorporating power-law spatial correlations through fractional Laplacian operators, providing explicit parameterization [...] Read more.
Neural field dynamics in the cerebral cortex exhibit complex spatiotemporal patterns inadequately captured by classical integer-order diffusion models that assume exponentially decaying spatial interactions. This study establishes a stochastic fractional FitzHugh–Nagumo framework incorporating power-law spatial correlations through fractional Laplacian operators, providing explicit parameterization of non-local cortical connectivity characteristics. The inverse problem of estimating fractional orders and model parameters from electroencephalographic data is addressed through multi-objective optimization with rigorous train–test validation. Systematic sensitivity analysis across the parameter space (αu,αv)∈[1.0,2.0]×[1.0,2.0] identifies optimal subdiffusive characteristics at αu=αv=1.5, corresponding to power-law spatial kernels C(x)∼|x|−1.5 consistent with anatomical connectivity measurements. The optimized model achieves out-of-sample performance R2=0.973 on held-out test data, approaching the measurement noise ceiling. While classical FitzHugh–Nagumo models achieve comparable test accuracy, the fractional framework provides enhanced interpretability through explicit spatial interaction parameterization. The fractional orders serve as quantitative biomarkers of cortical network organization, enabling data-driven characterization across brain states and neurological conditions. The methodology establishes computational foundations for clinical applications in epilepsy monitoring, neurodegenerative disease detection, and brain–computer interfaces. Full article
(This article belongs to the Special Issue Recent Advances in Fractal and Fractional Calculus)
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13 pages, 278 KB  
Article
Solving Fractional Differential Equations via New Relation-Theoretic Fuzzy Fixed Point Theorems
by Waleed M. Alfaqih, Salvatore Sessa, Hayel N. Saleh and Mohammad Imdad
Mathematics 2025, 13(16), 2582; https://doi.org/10.3390/math13162582 - 12 Aug 2025
Cited by 2 | Viewed by 711
Abstract
In this paper, we present the notion of fuzzy R−F−contractive mappings and provide some fuzzy fixed point results in the setting of fuzzy metric spaces, which are endowed with binary relations. Furthermore, we apply our newly established fuzzy fixed [...] Read more.
In this paper, we present the notion of fuzzy R−F−contractive mappings and provide some fuzzy fixed point results in the setting of fuzzy metric spaces, which are endowed with binary relations. Furthermore, we apply our newly established fuzzy fixed point results to solve certain boundary value problems for nonlinear fractional differential equations involving the Caputo fractional derivatives. Also, we provide some examples to show the utility of our new results. Full article
(This article belongs to the Special Issue Recent Advances in Fractal and Fractional Calculus)
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