Search Results (2,285)

In this paper, we investigate the finite-time consensus problem of a fractional-order multi-agent system with repetitive motion. The system under consideration consists of robotic agents with a leader and a fixed communication topology. A distributed open-closed-loop PDα fractional-order iterative learning control (FOILC) algorithm is proposed. The finite-time uniform convergence of the proposed algorithm is analyzed, and sufficient convergence conditions are derived. The theoretical analysis demonstrates that, as the number of iterations increases, each agent can achieve complete tracking within a finite time by appropriately selecting the gain matrices. Simulation results are presented to verify the effectiveness of the proposed method. Full article

Underground hydrogen storage (UHS) in depleted oil and gas reservoirs is regarded as a highly promising subsurface option due to its large storage capacity. In such reservoirs, the pore structure provides the primary space for hydrogen storage and governs matrix flow and diffusion. Tree-shaped fracture networks generated by hydraulic fracturing or cycling injection–production typically exhibit much higher transmissivity and serve as the dominant pathways. In this study, the geometry of multilevel branching fractures was parameterized, and two classes of tree-shaped fracture configurations were constructed, including point–line-type (PLTSF) and disc-shaped (DSTSF) networks. Analytical models were developed to evaluate the equivalent permeability of tree-shaped fracture networks with either elliptical or rectangular cross-sections. The Klinkenberg slip correction and a gas-type factor associated with molecular kinetic diameter were incorporated. The apparent equivalent permeability of hydrogen (k_app,H2) was quantified and compared with those of nitrogen and methane under identical conditions. The main findings were as follows: (1) the fracture width ratio (β) was identified as the primary factor controlling network conductivity, while the height ratio (α) amplified or attenuated this effect at a given β; (2) as the main-fracture aspect ratio, the branching order (n) or branching angle (θ) increased, the rectangular cross-sections were more favorable for maintaining higher permeability compared to the elliptical cross-section; (3) under typical operating pressures of 5–30 MPa, the apparent permeability of hydrogen was approximately 2–9% higher than that of methane and nitrogen; and (4) by introducing the fracture volume fraction, the REV-scale equivalent-permeability expression was derived for fractured rock masses containing tree-shaped fracture networks. The proposed framework provides a theoretical basis and parametric support for quantifying fracture flow capacity for UHS in depleted reservoirs. Full article

This paper concentrates on fractional-order fuzzy cellular neural networks (FOFCNNs) with bounded uncertainties and transmission delays. To better capture real-world dynamic behaviors, the fuzzy AND and OR operators are employed to construct drive-response systems. For the fixed-time synchronization task of the systems, a novel multi-module feedback controller incorporating three functional terms is designed. These terms aim to eliminate delay effects, ensure fixed-time convergence, and reduce parameter conservativeness. Leveraging the properties of fractional-order operators and our multi-module control scheme, new synchronization criteria of the studied drive-response systems can be established within a predefined time. An upper bound on the settling time is derived, depending on the system size and control parameters, but independent of the initial conditions. A significant corollary is derived for the case of no uncertainties under the nonlinear controller. Numerical experiments discuss the impact of uncertainties and delays on synchronization, and confirm the validity of the results presented in this study. Full article

In this article, we propose an expanded mixed finite element method for variable-coefficient fractional dispersion equations (FDEs). By introducing two intermediate variables, $p=Du$ and $\sigma =-I_{\theta }^{\beta }p$, the FDEs are reformulated into a mixed system involving only lower-order derivatives. Based on this, we construct an expanded mixed variational framework and prove the weak coercivity in the sense of the LBB condition over appropriately chosen Sobolev spaces, thereby ensuring the well-posedness of the formulation. Then, we develop an expanded mixed finite element scheme and prove that the unique expanded finite element solution possesses optimal approximation accuracy to the fractional flux $\sigma ,$ the gradient p and the unknown u. Finally, numerical experiments are conducted to verify the efficiency and accuracy of the proposed method. Full article

In this study, we address a coupled system of nonlinear fractional delay differential equations subject to cross-coupled multi-point boundary conditions. By utilizing the generalized power Caputo fractional derivative, we present a unified theoretical framework that encompasses several operators—including the Atangana–Baleanu, Caputo–Fabrizio, and weighted Hattaf derivatives—as special cases. This generality ensures that our results remain applicable across a broad family of fractional kernels. We transform the complex delay system into an equivalent integral form to derive sufficient criteria for the existence and uniqueness of solutions via fixed-point theory. Furthermore, we rigorously establish the Ulam–Hyers stability of the system, a critical property for ensuring robustness in the presence of perturbations. Finally, the theoretical findings are validated through a detailed numerical study employing a predictor–corrector scheme adapted for fractional delay systems. The simulations highlight the sensitivity of solutions to the memory kernel and fractional orders and include a systematic exploration of delay effects. Full article

Mock theta functions, introduced by Ramanujan in his last letter to Hardy, play a significant role in q-series theory and have natural connections to fractional q-calculus. In this paper, we study bilateral hypergeometric series of the form ${}_2{\psi }_2$= ${\sum }_{n=-\infty }^{\infty }{\displaystyle \frac{{(a,b;q)}_n}{{(c,d;q)}_n}}{z}^n,$ where ${(a;q)}_n$ denotes the q-shifted factorial. Using Slater’s three-term transformation formula for bilateral ${}_2{\psi }_2$ series, we derive new identities for Ramanujan’s mock theta functions of orders 2, 3, 6, and 8. These transformations reveal previously unknown relationships between different q-series representations and extend the classical theory of mock theta functions within the framework of q-special functions. Full article

This paper presents a reduced-order Legendre–Galerkin extrapolation (ROLGE) method combined with the scalar auxiliary variable (SAV) approach (ROLGE-SAV) to numerically solve the time-fractional Allen–Cahn equation (tFAC). First, the nonlinear term is linearized via the SAV method, and the linearized system derived from this SAV-based linearization is time-discretized using the shifted fractional trapezoidal rule (SFTR), resulting in a semi-discrete unconditionally stable scheme (SFTR-SAV). The scheme is then fully discretized by incorporating Legendre–Galerkin (LG) spatial discretization. To enhance computational efficiency, a proper orthogonal decomposition (POD) basis is constructed from a small set of snapshots of the fully discrete solutions on an initial short time interval. A reduced-order LG extrapolation SFTR-SAV model (ROLGE-SFTR-SAV) is then implemented over a subsequent extended time interval, thereby avoiding redundant computations. Theoretical analysis establishes the stability of the reduced-order scheme and provides its error estimates. Numerical experiments validate the effectiveness of the proposed method and the correctness of the theoretical results. Full article

This paper presents a novel high-order numerical method. The proposed scheme utilizes polynomial generating functions to achieve p order accuracy in time for the Grünwald–Letnikov fractional derivatives, while maintaining second-order spatial accuracy. By incorporating a short-memory principle, the method remains computationally efficient for long-time simulations. The authors rigorously analyze the stability of equilibrium points for the fractional vegetation–water model and perform a weakly nonlinear analysis to derive amplitude equations. Convergence analysis confirms the scheme’s consistency, stability, and convergence. Numerical simulations demonstrate the method’s effectiveness in exploring how different fractional derivative orders influence system dynamics and pattern formation, providing a robust tool for studying complex fractional systems in theoretical ecology. Full article

This research work investigates the existence, uniqueness, and stability of solution for a time-fractional fourth-order partial differential equation, subject to two initial conditions and four nonlocal integral boundary conditions. The equation incorporates several key components: the Caputo fractional derivative operator, the Laplace operator, the biharmonic operator, as well as terms representing frictional and viscoelastic damping. The presence of these elements, particularly the nonlocal boundary constraints, introduces new mathematical challenges that require the development of advanced analytical methods. To address these challenges, we construct a functional analytic framework based on Sobolev spaces and employ energy estimates to rigorously prove the well-posedness of the problem. Full article

This work explores the qualitative dynamics of the modified unstable time-fractional nonlinear Schrödinger equation (mUNLSE), a model applicable to nonlinear wave propagation in plasma and optical fiber media. By transforming the governing equation into a planar conservative Hamiltonian system, a detailed bifurcation study is carried out, and the associated equilibrium points are classified using Lagrange’s theorem and phase-plane analysis. A family of exact wave solutions is then constructed in terms of both trigonometric and Jacobi elliptic functions, with solitary, kink/anti-kink, periodic, and super-periodic profiles emerging under suitable parameter regimes and linked directly to the type of the phase plane orbits. The validity of the solutions is discussed through the degeneracy property which is equivalent to the transmission between the phase orbits. The influence of the fractional derivative order on amplitude, localization, and dispersion is illustrated through graphical simulations, exploring the memory impacts in the wave evolution. In addition, an externally periodic force is allowed to act on the mUNLSE model, which is reduced to a perturbed non-autonomous dynamical system. The response to periodic driving is examined, showing transitions from periodic motion to quasi-periodic and chaotic regimes, which are further confirmed by Lyapunov exponent calculations. These findings deepen the theoretical understanding of fractional Schrödinger-type models and offer new insight into complex nonlinear wave phenomena in plasma physics and optical fiber systems. Full article

This study explores finite-time synchronization (FTS) in fractional-order quaternion-valued neural networks (FQVNNs) characterized by discontinuous activation functions and uncertainties in parameters. Initially, leveraging the properties of the Mittag-Leffler function along with fractional-order (F-O) delayed differential inequalities, a novel finite-time stability theorem for F-O systems is established, building upon previous research findings. Next, based on norm definitions, two state feedback controllers employing quaternion 1-norm and quaternion 2-norm are devised to ensure FTS for the system under consideration. Following this, by utilizing differential inclusion theory, examining the quaternion sign function, employing advanced inequality methods, applying principles of F-O differential equations, and using the Lyapunov functional approach, new criteria for achieving FTS in FQVNNs are formulated. Additionally, precise estimates for the settling time are presented. In conclusion, two carefully designed numerical examples are included to corroborate the theoretical results derived. Full article

In the current study, we used Chebyshev’s Pseudospectral Method (CPM), a novel numerical technique, to solve nonlinear third-order Emden–Fowler delay differential (EF-DD) equations numerically. Fractional derivatives are defined by the Caputo operator. These kinds of equations are transformed to the linear or nonlinear algebraic equations by the proposed approach. The numerical outcomes demonstrate the precision and efficiency of the suggested approach. The error analysis shows that the current method is more accurate than any other numerical method currently available. The computational analysis fully confirms the compatibility of the suggested strategy, as demonstrated by a few numerical examples. We present the outcome of the offered method in tables form, which confirms the appropriateness at each point. Additionally, the outcomes of the offered method at various non-integer orders are investigated, demonstrating that the result approaches closer to the accurate solution as a value approaches from non-integer order to an integer order. Additionally, the current study proves some helpful theorems about the convergence and error analysis related to the aforementioned technique. A suggested algorithm can effectively be used to solve other physical issues. Full article

This paper considers the solution behavior and dynamical properties of the variable-order fractional Newton–Leipnik system defined via Liouville–Caputo derivatives of variable order. In contrast to integer-order models, the presence of variable-order fractional operators in the Newton–Leipnik structure enriches the model by providing memory-dependent effects that vary with time; hence, it is capable of a broader and more flexible range of nonlinear responses. Numerical simulations have been conducted to study how different order functions influence the trajectory and qualitative dynamics: clear transitions in oscillatory patterns have been identified by phase portraits, time-series profiles, and three-dimensional state evolution. The work goes further by considering the development of bifurcations and chaotic regimes and stability shifts and confirms the occurrence of several phenomena unattainable in fixed-order and/or integer-order formulations. Analysis of Lyapunov exponents confirms strong sensitivity to the initial conditions and further details how the memory effects either reinforce or prevent chaotic oscillations according to the type of order function. The results, in fact, show that the variable-order fractional Newton–Leipnik framework allows for more expressive and realistic modeling of complex nonlinear phenomena and points out the crucial role played by evolving memory in controlling how the system moves between periodic, quasi-periodic, and chaotic states. Full article

Cryotherapy and radiofrequency (RF) treatments modulate tissue temperature to induce therapeutic effects; however, improper application can result in thermal injury. Traditional temperature-based monitoring methods rely on multiple thermal sensors whose accuracy strongly depends on their number and spatial positioning, often failing to detect early tissue crystallization. This study introduces a fractional order bioimpedance modelling framework for the early detection of tissue freezing during cryogenic and thermal medical treatments, with the feasibility and effectiveness of this approach having been reported in our prior publications. While bioimpedance spectroscopy itself is a well-est. The corresponablished technique in biomedical engineering, its novel application to predict and identify premature freezing events provides a new pathway for safe and efficient energy-based therapies. Fractional-order models derived from the Cole family accurately reproduce the complex electrical behavior of biological tissues using fewer parameters than classical integer-order models, thus reducing both hardware requirements and computational cost. Experimental impedance data from human abdominal, gluteal, and femoral regions were modelled to extract fractional parameters that serve as sensitive indicators of phase-transition onset. The results demonstrate that the proposed approach enables real-time identification of freezing-induced electrical transitions, offering a physiologically grounded alternative to conventional temperature-based monitoring. Furthermore, the fractional order bioimpedance method exhibits high reproducibility and selectivity, and its analytical figures of merit, including the limits of detection and quantification, support its use for reliable real-time tissue monitoring and early injury detection. Overall, the proposed fractional order bioimpedance framework enhances both safety and control precision in cryogenic and thermal medical applications. Full article

The energy efficiency and hover stability of unmanned aerial vehicles are critical factors, since improper battery utilization and unstable control are major sources of operational failures and accidents. The proportional–integral–derivative (PID) controller, which is applied in approximately 97% of multirotor unmanned aerial vehicle (UAV) systems, is widely used due to its simplicity; however, it is sensitive to external disturbances and often fails to ensure optimal energy utilization, resulting in reduced flight time. Therefore, the experimental investigation of advanced control methods in a real physical environment is well justified. The objective of the present research is the comparative evaluation of seven control strategies—PID, linear quadratic controller with integral action (LQI), model predictive control (MPC), sliding mode control (SMC), backstepping control, fractional-order PID (FOPID), and H∞ control—using a single-degree-of-freedom drone test platform in a MATLAB R2023b-Arduino hardware-in-the-loop (HIL) environment. Although the theoretical advantages and model-based results of the aforementioned control methods are well documented, the number of real-time comparative HIL experiments conducted under identical physical conditions remains limited. Consequently, only a small amount of unified and directly comparable experimental data is available regarding the performance of different controllers. The measurements were performed at a reference height of 120 mm under disturbance-free conditions and under wind loading with a velocity of 10 km/h applied at an angle of 45°. The controller performance was evaluated based on hover accuracy, settling time, overshoot, and real-time measured power consumption. The results indicate that modern control strategies provide significantly improved energy efficiency and faster stabilization compared to the PID controller in both disturbance-free and wind-loaded test scenarios. The investigations confirm that several advanced controllers can be applied more effectively than the PID controller to enhance hover stability and reduce energy consumption. Full article