Keywords: Degree of nondensifiability, measure of noncompactness, fixed point, topological space, space-filling curves, continuum, alpha-dense curves, fractional differential equations

Keywords: Mathematical methods, History of Fractional-Order Calculus, Others
Abstract: The Erdélyi-Kober fractional integrals and derivatives are variants of the Riemann-Lioville with very wide applications due to the freedom to chose their three arbitrary parameters. In this paper we introduce and study a generalization of the Erdélyi-Kober operators of fractional integration where the elementary kernel-function is replaced by a suitably chosen I_{1,1}^{1,0}-function. The I-functions are generalized hypergeometric functions, introduced by Rathie in 1997 as extensions of the Fox H-functions and Meijer G-functions, but have been not popular because of their rather complicated structure and multi-valued behaviour. However, it happens that they are useful not only in statistical physics, but include also important special functions in mathematics. In our previous works we related the I-functions to some specific special functions of fractional calculus, among which analogues of the Mittag-Leffler and Le Roy type functions. Here, we propose a theory of an I-function generalization of the Erdélyi-Kober fractional integrals, that will serve further as a base for further extension of the generalized fractional calculus with operators of fractional multi-order. Some preliminary results were presented also in our recent paper [8].

Keywords: Mathematical methods, History of Fractional-Order Calculus, Others
Abstract: Till recently, the operators of Generalized Fractional Calculus (GFC) with the Fox H- and Meijer G-functions as singular kernels, have been considered as the most general case of integrations and differentiations of fractional orders and fractional multi-orders. The H-function (as an extension of the G-function) is a generalized hypergeometric function including practically almost all Special Functions (SF) related to classical Calculus and to fractional order calculus (FC). However, it happened that still there are important classes of SF that cannot be represented in terms of the H-function, like the polylogarithms, Riemann Zeta-function, Feynman integrals, Le Roy function and their recent multi-index versions. That is why, the so-called Rathie I-function has attracted our attention as a more general analogue of the H-function, since it encompasses also these other classes of special functions. Now, we introduce GFC operators involving as kernels the I-function, and prove that they satisfy all the basic axioms of FC, extend the GFC with H- and G-functions [3] and provide relations to new classes of SF. It happens that such SF, for example the multi-index Le Roy type functions, appear as eigen functions for the new GFC operators.

Keywords: Mathematical methods, Physics, Singularities Analysis and Integral Representations
Abstract: This paper proposes an efficient numerical algorithm for the time-space fractional diffusion equation with temporal Caputo-Hadamard derivative and spatial fractional Laplacian in two dimensions. For the time discretization, the spectral deferred correction (SDC) method is utilized. In the mean while, the fractional centered finite difference formula is adopted to approximate the two-dimensional fractional Laplacian. Numerical example is presented to illustrate the propose scheme.

Keywords: History of Fractional-Order Calculus, Mathematical methods, Others
Abstract: The document presents a class of quasi-fractional analytic functions, exploring their properties in complex analysis and fractional calculus. Definitions, theorems, and proofs are established that link these functions with concepts such as Gauss’s Theorem and the Cauchy- Pompeiu formula. Additionally, the relationship between harmonicity and quasi-fractional analyticity is investigated, also introducing the concept of quasi-generalized fractional analytic functions.

Keywords: Automatic Control & Stability, Artificial Intelligence, Electrical Engineering & Electromagnetism
Abstract: Power electronic systems often suffer from unmodeled dynamics, parameter variations, and nonlinearities, which can affect closed-loop stability and transient performance. This paper proposes a novel metaheuristic approach—the PID-Based Search Algorithm (PSA)—to robustly design feedback controllers, with a focus on DC-DC buck converters. A fractional-order PI–PD (FOPI–FOPD) cascade controller is introduced, leveraging fractional calculus to extend conventional control actions via non-integer orders λ and μ, thus enhancing design flexibility and robustness. PSA simultaneously tunes both gains and fractional orders by minimizing the Integral of Time-weighted Absolute Error (ITAE), optimizing both transient and steady-state behavior. Time-domain simulations demonstrate that the PSA-optimized FOPI–FOPD controller outperforms its integer-order counterpart in terms of settling time, overshoot, and disturbance rejection, confirming its correctness and effectiveness in nonlinear converter applications.

Keywords: Automatic Control & Stability, Biology & Biomedicine, System identification & Modeling
Abstract: This article presents a new modeling approach of the blood glucose system using fractional-order operators, and the design and implementation of an indirect model reference adaptive controller (MRAC) for an artificial pancreas, aimed at managing the human glycaemic system. The proposed fractional-order adaptive control strategy is developed to dynamically adjust to the individual variations in the metabolic processes, providing a personalized methodology to glucose regulation. A comparative study with the standard indirect adaptive controller is provided to illustrate the superiority of the proposed modeling and control scheme.

Keywords: Biology & Biomedicine, Automatic Control & Stability, System Analysis & Dynamics
Abstract: Delivering safe, patient‑specific anesthesia demands simultaneous regulation of general‑anesthesia and hemodynamic. This study proposes a steady state decoupled Fractional-Order Model Predictive Control (FOMPC) design of hemodynamic system under general anesthesia conditions particularly for patients during surgical procedures. We used a patient numerical model to validate the introduced control algorithm with comparative tests against other controllers. The aim is to stabilize the patient's mean arterial pressure (MAP) and cardiac output (CO), taking into account the subsystems coupling (synergic and antagonistic). Based on a fractional-order predictive model we show via intensive simulation experiments the robustness of the system against internal interactions and external disturbances and its efficiency to meet the clinical objectives.

Keywords: Biology & Biomedicine, System Analysis & Dynamics, Others
Abstract: The removal of dyes from waste water has become a major environmental concern due to their persistence and potential toxicity, among advanced oxidation techniques such as photo catalysis Unlike conventional kinetic models pseudo first order or pseudo second order, fractional order kinetics provides a better description of actual mechanisms involved in these reaction. The photo catalysis used in this study was synthesized in the laboratory. It was characterized by several physicochemical methods: XRD analysis to confirm the formation of phases, SEM to determine the calcination temperature, FTIR to identify functional groups and EDS analysis to verify the atom present.

Keywords: Biology & Biomedicine, Special Functions
Abstract: Diffusion in porous media, including biological tissues, often deviates from classical Fickian behavior and can be more parsimoniously described by space-fractional diffusion models. The study analyzes a reaction–diffusion system in two concentric compartments: a proximal, finite-radius region containing a source, and an unbounded outer region without a source but with first-order decay of the diffusing species, capturing the foreign body response around an implanted electrode. Microscale tissue heterogeneity is represented by a space-fractional Riesz Laplacian operator acting on the concentration, providing a flexible framework for estimating transport parameters from experimental data. The steady state is obtained via Hankel and Mellin transforms, yielding a Fox H-function representation; in the integer-order limit, the solution reduces to a superposition of modified Bessel functions of the first and second kinds. Solutions are computed by numerically evaluating the underlying Bessel integral expressions using an accelerated double exponential quadrature method.

Keywords: Robotics, Automatic Control & Stability, System Analysis & Dynamics
Abstract: This paper presents a novel Fractional-Order Model Reference Adaptive Control (FO-MRAC) strategy for the adaptive feedback linearization of a nonlinear Delta parallel robot. By leveraging fractional calculus, the proposed approach enhances both robustness and adaptability in the presence of model uncertainties. The fractional-order framework enables a more accurate representation of the robot's dynamics by capturing memory effects and viscoelastic behavior phenomena typically neglected in integer-order models. The adaptive control mechanism continuously adjusts the controller parameters in real time, ensuring precise trajectory tracking without requiring exact knowledge of the system model.

Keywords: Automatic Control & Stability
Abstract: This paper aims to design and implement a fractional-order PID controller optimized using the Particle Swarm Optimization (PSO) algorithm for the motion control of a differential-drive mobile trolley (DDMT). The proposed approach seeks to enhance the system’s dynamic response and tracking performance by leveraging the flexibility of fractional calculus and the global search capabilities of metaheuristic optimization. To assess the effectiveness of the proposed controller, a comparative study is conducted involving three control strategies: the conventional PID controller, a non-optimized FOPID controller, and a PSO-optimized FOPID controller. The performance is evaluated based on several criteria, including accuracy, stability, response time, and the Integral Squared Error (ISE). Simulation results demonstrate that the PSO-optimized FOPID controller outperforms the other approaches across all metrics. These findings highlight the advantages of fractional-order control combined with intelligent optimization techniques for complex mobile trolley systems.

Keywords: Electrical Engineering & Electromagnetism, Automatic Control & Stability, Mechatronics
Abstract: Fractional calculus is the most popular form of control engineering in many fields, including electric drive ‎applications. One of the most common applications in all fields of electric drives is the control of the dual-star ‎induction machine (DSIM). Several control techniques have been proposed for this type of multiphase motor, ‎ranging from classical PID-based methods to the most sophisticated advanced methods, including fractional-‎order controllers. Furthermore, in the case of a DSIM, which has been widely studied recently, inevitable ‎harmonic currents are generated, which is a major problem and leads to increased losses and reduced system ‎efficiency. Therefore, this paper presents a fractional order controller optimized using a dedicated method to ‎improve system performance while minimizing overshoot, reducing response time, and minimizing rejection ‎time, and a modified switching table has been developed to reduce harmonic currents using virtual voltage ‎vectors. Simulation results validate the effectiveness of the proposed method.‎

Keywords: Automatic Control & Stability, Robotics, Electrical Engineering & Electromagnetism
Abstract: Quadrotor UAVs demand robust control strategies to handle inherent nonlinearities, model uncertainties, and external disturbances. While conventional PID control is widely used due to its simplicity, its performance degrades under dynamic perturbations and aggressive maneuvers. This paper presents an enhanced Active Disturbance Rejection Control (ADRC) framework for quadrotors, leveraging a fractional-order cascaded nonlinear Extended State Observer (ESO) to improve disturbance estimation and rejection. The proposed ESO combines fractional calculus for precise error dynamics tuning with a cascaded structure to decouple and compensate for multi-source disturbances (e.g., wind gusts, payload variations, and unmodeled dynamics). Comparative flight simulations demonstrate that the fractional cascaded nonlinear ESO-ADRC achieves superior trajectory tracking and disturbance rejection compared to the conventional linear ESO-ADRC, particularly under high-speed flight and sudden disturbances. The results validate the observer’s enhanced robustness and adaptability, offering a promising solution for autonomous quadrotor control in complex environments.

Keywords: Automatic Control & Stability, Mathematical methods, System Analysis & Dynamics
Abstract: In this paper we extend the classical high gain theorem to linear fractional systems of commensurate order. The main purpose of the present work is to stabilize this special class of fractional order systems using a high gain output feedback. The basis for extending this theorem lies in the stability results derived by Matignon. The derived theorem ensures the stabilization of linear fractional systems of arbitrary commensurate order under certain assumptions. A numerical example illustrates the effectiveness of the approach. Furthermore, We employ the Particle Swarm Optimization (PSO) algorithm to tune the high gain feedback so that the output of the closed-loop system reaches a desired value. The study provides valuable insights for linear fractional order systems of commensurate order.

Keywords: Automatic Control & Stability, Robotics
Abstract: This paper proposes an enhanced cascade control approach for trajectory tracking of nonholonomic mobile robots. The proposed method integrates a classical kinematic controller with a novel Fractional-Order Finite-Time Synergetic Controller (FOFTSC). Unlike classical synergetic controllers that offer asymptotic convergence, the proposed FOFTSC achieves finite-time error convergence and exhibits superior transient performance and robustness against external disturbances. A comparative study with a conventional cascade control strategy, that combines a Lyapunov-based kinematic controller and a classical synergetic dynamic controller, shows the performance superiority of the proposed FOFTSC. Simulation results across multiple trajectory scenarios demonstrate that our method, which features a fractional-order finite-time synergetic controller, achieves significantly faster convergence and enhanced tracking accuracy, particularly in the presence of dynamic disturbances.