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Advances in Fractional Differential Equations and Applications
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Advances in Fractional Differential Equations and Applications

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 September 2026 | Viewed by 1005

Editors

Dr. Farzana A. McRae

E-Mail Website
Guest Editor
Department of Mathematics, Catholic University of America, Washington, DC 20064, USA
Interests: ordinary differential equations; fractional integral equations; impulsive differential equations; integral equations
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Zahia Drici

E-Mail Website
Guest Editor
Department of Mathematics, Illinois Wesleyan University, Bloomington, IL, USA
Interests: ordinary differential equations; dynamical systems; difference and functional equations; integral equations; electromagnetic theory and its applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The Special Issue, “Advances in Fractional Differential Equations and Applications”, aims to present recent theoretical and applied advances in the field of fractional differential equations. This issue focuses on the development of fractional calculus and fractional-order models, including analytical, qualitative, and numerical approaches to fractional differential equations.

Topics of interest include, but are not limited to, the following:

  • Theory of fractional differential equations;
  • Existence, uniqueness and stability of solutions;
  • Analytical methods;
  • Numerical and computational techniques;
  • Fractional-order dynamical systems and control;
  • Applications in physics, engineering, biology, finance, and other applied sciences;
  • Modelling of nonlocal and memory-dependent phenomena.

This Special Issue welcomes original research articles and high-quality review papers that contribute to the advancement of fractional differential equations and their applications within the scope of mathematics and related disciplines.

Dr. Farzana A. McRae
Prof. Dr. Zahia Drici
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-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 derivative
  • Caputo fractional derivative
  • Riemann–Liouville fractional derivative
  • fractional differential equations
  • fractional-order systems
  • nonlocal operators
  • fractional-order control
  • non-integer order modelling
  • analytical methods
  • numerical methods
  • stability analysis
  • mathematical modelling

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Published Papers (2 papers)

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45 pages, 1618 KB  
Article
Convergence-to-Zero and Guaranteed-Cost Synchronization of Caputo–Hadamard Fractional-Order Systems with a Time-Varying Delay
by Ymnah Alruwaily, Slim Dhahri and Foued Mtiri
Mathematics 2026, 14(15), 2788; https://doi.org/10.3390/math14152788 - 4 Aug 2026
Viewed by 333
Abstract
In this paper, the convergence-to-zero and finite-horizon guaranteed-cost synchronization criteria are developed for linear Caputo–Hadamard fractional-order systems with an admissible time-varying delay. The delay assumption is expressed in such a way that is consistent with the Caputo–Hadamard Halanay inequality and the memory structure [...] Read more.
In this paper, the convergence-to-zero and finite-horizon guaranteed-cost synchronization criteria are developed for linear Caputo–Hadamard fractional-order systems with an admissible time-varying delay. The delay assumption is expressed in such a way that is consistent with the Caputo–Hadamard Halanay inequality and the memory structure of the model, which is logarithmic in time. This analysis includes a quadratic Caputo–Hadamard Lyapunov method, Schur-complement bounds for the delayed channel and a supremum argument in logarithmic time. An important novelty in the proposed approach is that the current state, the delayed state and the Caputo–Hadamard derivative are not considered as independent augmented variables; this prevents the structural feasibility obstacle from occurring when using full-space residual LMI formulations. The convergence-to-zero condition is first established for the drive system. Next, a fixed-gain guaranteed-cost synchronization theorem is established and, by using a standard change of variables, a convex controller-synthesis condition is arrived at. An explicit logarithmic-time form of the finite-horizon cost estimate is derived. The criteria are further extended to systems with several admissible delays and to systems with norm-bounded parametric uncertainty. Four numerical examples are reported, in which feasible matrices, the controller gain, strict eigenvalue margins and a comparison of the simulated cost and the theoretical upper bound are given, together with a quantitative comparison against augmented-state linear matrix inequality formulations, a scalability study up to a dimension of 30 and a sensitivity study. The simulations show the dynamics that the theory predicts; the convergence to the asymptotics is valid for the LMI certificates checked in the simulations and for the Caputo–Hadamard Halanay inequality. In conclusion, the paper delivers a complete and numerically verifiable design chain for Caputo–Hadamard synchronization: admissibility of a possibly unbounded time-varying delay is checked directly, a stabilizing gain is obtained from a convex program whose largest block has size 2n instead of 3n, and an a priori cost certificate JT* is produced from the same feasible variables; on the reported benchmark, the method retains 95.7% of the admissible delay-channel gain of an augmented-state formulation while solving up to 21 times faster at dimension 30. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Equations and Applications)
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13 pages, 277 KB  
Article
On the Mild Solutions of Second-Order Θ-Caputo Fractional Boundary Value Problems
by Mouataz Billah Mesmouli, Abdelouaheb Ardjouni, Loredana Florentina Iambor and Taher S. Hassan
Mathematics 2026, 14(9), 1434; https://doi.org/10.3390/math14091434 - 24 Apr 2026
Viewed by 335
Abstract
In this paper, we study a class of second-order fractional boundary value problems involving Θ-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the Θ-fractional framework. Existence results are [...] Read more.
In this paper, we study a class of second-order fractional boundary value problems involving Θ-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the Θ-fractional framework. Existence results are obtained via Krasnoselskii’s fixed point theorem, while uniqueness is established using the Banach contraction principle under suitable Lipschitz-type conditions. The obtained results extend several earlier works on Caputo, Hadamard–Caputo, and Riemann–Liouville fractional derivatives. Two examples are presented to illustrate the applicability of the theoretical results. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Equations and Applications)
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