Search Results (875)

Fractional optimal control problems provide an effective mathematical framework for modeling dynamical systems with memory, hereditary behavior, and anomalous diffusion effects. However, the nonlocal nature of Caputo fractional operators and the reduced regularity of fractional solutions pose significant challenges for the development of accurate and efficient computational methods. In this paper, we develop a spectral-fractional Physics-Informed Neural Network (Spectral-fPINN) framework for solving fractional optimal control problems governed by Caputo fractional differential equations. The proposed methodology combines normalized shifted Legendre spectral approximations, fractional operational matrix formulations, and physics-informed optimization within a unified computational framework. Unlike conventional PINN and fPINN approaches, which directly approximate the unknown solution variables, the proposed framework predicts the spectral coefficient vectors associated with the shifted Legendre basis functions, yielding a low-dimensional global representation with improved approximation efficiency. Caputo fractional derivatives are evaluated through spectral operational matrices, while the resulting optimization problem is discretized using Gauss–Legendre quadrature and solved through gradient-based optimization. In addition, a theoretical analysis of the proposed Spectral-fPINN framework is presented, including approximation, consistency, stability, and convergence results, together with error estimates and residual control properties. Several benchmark linear and nonlinear fractional optimal control problems are investigated to validate the proposed methodology. The numerical results demonstrate excellent agreement with exact solutions, very small residual errors, and rapid spectral coefficient decay, confirming the high-order accuracy and robustness of the proposed approach. Overall, the proposed Spectral-fPINN framework provides an accurate, stable, and computationally efficient methodology for solving a broad class of fractional optimal control problems. Full article

Rooftop rainwater harvesting (RWH) offers a practical adaptation option for Mediterranean cities where water scarcity is amplified by seasonal rainfall and climate variability. This study reports early findings from a simplified monthly water balance screening model for a typical residential building, driven by ERA5-Land monthly precipitation for Antalya and İzmir (Türkiye). Scenarios cover roof areas of 250–3000 m² and practical tank capacities of 2–100 m³ under a fixed non-potable demand of 0.20 m³/day. The model tracks monthly storage dynamics and supply demand in order to compute demand coverage and monthly reliability (i.e., fraction of months in which full demand is met). Reliability-based storage thresholds (≥0.80) are derived for four evaluation windows (1996–2010, 2011–2025, 1996–2025, 1950–2025) to explore climate sensitivity. In parallel, a guideline-style sizing which is consistent with the Turkish rainwater harvesting guideline is implemented using a three-day storage rule based on the wettest month potential. To enable a like-for-like comparison, the collection losses are harmonized by setting loss to 0.10 in the simulation and efficiency to 0.90 in the guideline method. The results show stable thresholds for Antalya but stronger period sensitivity in İzmir. They also quantify cases where guideline sizing does not achieve the target reliability under dry season constraints. This approach supports the rapid, climate-aware pre-design of small- to medium-scale urban RWH systems. Full article

Aquatic and semi-aquatic plant assemblages, water quality, riparian habitat, and landscape conditions were assessed for 140 ponds located in the Ozarks region in Arkansas and Missouri in order to better describe their occurrences and distributional patterns. Local environmental and landscape-level determinants that shape their diversity and influence their respective distributions, particularly in light of urbanization, were also assessed. Ozark ponds are highly variable in terms of physical structure, habitat quality, and plant diversity. Urban ponds were generally of lower quality in terms of environmental attributes compared to those in non-urban areas, but they had similar plant taxa richness as well as numbers of non-native species compared to their non-urban counterparts. Ponds had high plant diversity (N = 204 taxa, $\overline{\mathrm{x}}$ = 9.89, range = 0–33). Taxa richness increased with increasing pond size, and urban ponds had slightly more species on average compared to non-urban ponds (10.38 vs. 9.58, respectively). Spatial beta diversity of plants showed a high dissimilarity among ponds, with turnover being the dominant fraction. Beta diversity also followed a significant distance-decay model. These findings show that urban Ozark ponds serve as important habitats for a broad variety of aquatic plants. Full article

In this study, a detailed analytical-numerical study of the complex modified Korteweg–De Vries (mKdV) model with truncated M-fractional derivative is carried out to investigate the effects of the fractional order on nonlinear wave propagation. The fractional partial differential equation is solved by an appropriate fractional traveling wave transformation, which transforms it into a nonlinear ordinary differential equation. Two very powerful analytical methods are then used: the modified sub-equation method and the Kumar–Malik method, which give the exact closed-form solutions. The obtained semi-analytical numerical approximations are then obtained from the Differential Transformation Method (DTM). Bright and dark solitons, kink-type waves, periodic and rational solutions, exponential solutions, and Jacobi elliptic functions are found for a variety of parametric regimes. Explicit compatibility conditions and parametric constraints, which control the amplitude, width, and propagation, are derived. The DTM approximations are found to converge to the exact solutions with good accuracy, and the absolute errors are almost negligible, which validates the accuracy of the approximations and reliability of the solution. The three-dimensional visualizations of surface plots, two-dimensional profiles, and contour visualization further illustrate the dispersive dynamics and stability properties. Significance: This study shows that the truncated M-fractional derivative is a good operator to model memory-dependent nonlinear wave propagation. A new precise solution and reliable validation methods have been obtained for high-dimensional fractional nonlinear evolution equations in the hybrid analytical-numerical framework, which can be useful in plasma physics, nonlinear optics, and complex media. The present study contains restrictions for constant coefficients, a specific parametric regime, one fractional derivative definition, and experimental validation is not included. Future directions are limitations on constant coefficients, specific parametric regimes, one fractional derivative definition, and experimental validation is not included. The approach is to be extended in the future to variable coefficients, other fractional operators (Caputo, Riemann–Liouville), and to higher-order nonlinearities, and then to be experimentally tested in optical or plasma systems. Full article

Based on the Infinite State Representation (ISR) of the Riemann–Liouville integral, the energy stored in a fractional-order integrator is revisited, together with the energy dissipated through Joule losses. Using an idealized ultracapacitor model based on the Constant Phase Element (CPE), i.e., a fractional-order capacitor, theoretical expressions for the stored and dissipated energies during current charging of the CPE are derived. Numerical simulation of the fractional integrator over a frequency interval {ω_min, ω_max} validates a realistic CPE model, in which low-frequency modes correspond to energy storage, while high-frequency modes account for self-discharge and the origin of dissipated energy. This theoretical study leads to the definition of a new ultracapacitor model composed of an internal resistor and the previous realistic CPE, whose frequency-distributed representation enables prediction of the internal state variables and, consequently, the State of Charge. Full article

Nickel ferrite (NiFe₂O₄) is one of the most studied inverse-spinel ferrites because it combines moderate saturation magnetization, comparatively high electrical resistivity, chemical stability, and broad synthesis flexibility. Yet the literature shows that the measured structure and magnetism of NiFe₂O₄ are not intrinsic constants; they evolve strongly with dimensionality, size, thickness, strain state, cation distribution, surface spin disorder, and synthesis pathway. This review develops a unified theoretical and literature-based interpretation of how dimensionality reshapes the structural and magnetic behavior of NiFe₂O₄ across bulk ceramics, nanoparticles, one-dimensional nanostructures, polycrystalline thin films, and ultrathin epitaxial films. The review is anchored in the two uploaded nickel ferrite attachments and expanded using internet-sourced journal literature on spinel inversion, surface effects, mechanochemical synthesis, sputtered and pulsed laser deposited thin films, and epitaxial ultrathin-film anomalies. The central novelty of this article is the formulation of a dimensionality-dependent framework in which the observed magnetic response is governed by a competition among three coupled factors: (i) the cation-distribution function, which controls the A–B superexchange balance and therefore the net ferrimagnetic moment; (ii) the microstructural coherence function, which measures how crystallinity, strain, defects, and anti-phase boundaries preserve or degrade exchange continuity; and (iii) the surface/interface spin-order parameter, which quantifies the loss or reconfiguration of magnetic order at free surfaces and buried interfaces. Within this framework, bulk NiFe₂O₄ behaves as a near-equilibrium inverse spinel with relatively stable magnetization, whereas nanoscale NiFe₂O₄ experiences strong spin canting and finite-size suppression due to the growing fraction of disordered surface spins. Thin films introduce a distinct regime in which strain, texture, anti-phase boundaries, substrate mismatch, and growth kinetics determine both anisotropy and magnetization. In ultrathin epitaxial films, off-equilibrium cation redistribution and interface-controlled electronic reconstruction may even generate magnetization values far above bulk expectations. The review also compares major synthesis routes—solid-state reaction, sol–gel, co-precipitation, hydrothermal growth, reactive milling, combustion, pulsed laser deposition, and radio-frequency sputtering—and explains why each route biases the final dimensionality-dependent properties differently. A set of word-style equations is provided to formalize spinel inversion, finite-size suppression, anisotropy scaling, coercivity trends, and superparamagnetic crossover. Beyond summarizing the field, the review proposes a regime map linking dimensionality to characteristic structural defects and magnetic signatures, and it identifies unresolved questions concerning the true origin of enhanced magnetization in ultrathin NiFe₂O₄, the interplay between anti-phase boundaries and strain, and the distinction between intrinsic inversion changes and extrinsic substrate artifacts. The resulting article offers a submission-ready, originality-focused review that positions dimensionality as the master variable governing structure–magnetism correlations in nickel ferrite. Full article

According to data from the World Federation of Societies of Anesthesiologists, numerous countries across Asia and Africa have fewer than one anaesthesiologist per 100,000 people. Upskilling nurse anaesthetists in these regions is critical to improving clinical outcomes, and interactive virtual patient simulators offer a safe environment to explore complex clinical scenarios. This paper introduces an advanced general anaesthesia patient simulator engineered to bridge the accessibility gap left by existing platforms, which often require expert programming knowledge and restrict users to manual titration. Our simulator features an intuitive graphical user interface optimised for clinical education and natively supports both manual and closed-loop anaesthesia administration. The platform includes a suite of pre-designed controllers, specifically standard PIDs and two distinct fractional-order FO-PID variants, highlighting a novel robust FO-PID framework engineered to mitigate high patient variability. The deployment of these embedded controllers is demonstrated via a Depth of Hypnosis regulation case study and validated across a diverse cohort of 19 virtual patients. Closed-loop evaluation reveals that while the standard PID achieves a lower average mean squared error during the maintenance phase, the fractional-order alternatives deliver significantly superior robustness and inter-patient consistency. Ultimately, integrating this simulator into clinical training frameworks offers a viable pathway to reduce nursing workload and enhance patient safety through optimised automated drug delivery. Full article

This study investigates a 2D fractional order generalized thermoelastic problem in a homogeneous and isotropic thermoelastic hollow sphere. The sphere is exposed to a decaying heat source, and the governing equations are derived using a refined fractional-order Lord–Shulman (LS) model of generalized thermoelasticity. The Laplace transform technique is used to convert time-dependent PDEs into simpler ODEs in the Laplace domain. Its numerical inversion method is used to revert to the time domain. Numerical simulations are carried out to investigate the distributions of temperature, displacement, and stress fields within the hollow sphere. The obtained results reveal that both the fractional-order parameter and the variable thermal conductivity strongly affect the thermoelastic response, particularly the propagation characteristics of thermal waves, stress intensity, and relaxation behavior. In addition, the curvature of the hollow geometry plays an important role in modifying the radial and circumferential stress distributions and their attenuation throughout the medium. Full article

Arnold tongues are wedge-shaped regions in parameter space associated with mode locking and synchronization phenomena in nonlinear dynamical systems. The Caputo fractional standard map extends the classical standard map by incorporating long-memory effects through fractional derivatives and is known to generate Arnold tongue structures as the fractionality parameter approaches unity. In this paper, we investigate the fractional standard map applied to matrix-valued state variables, with particular emphasis on systems governed by high-dimensional nilpotent matrices. We show that the interplay between fractional memory and nilpotent algebra produces hierarchical families of Arnold tongues associated with divergent dynamics. This phenomenon is not observed in either the classical standard map or the non-fractional standard map of nilpotent matrices alone. For idempotent matrices, the fractional standard map retains the same level of dynamical complexity as its scalar counterpart. For nilpotent matrices, higher-order terms induce coupling between the map coefficients, giving rise to substantially richer dynamical behavior. This combination of fractional memory and nilpotent algebra provides a systematic framework for studying higher-dimensional nonlinear dynamics beyond the scalar setting. To support numerical investigations, an efficient computational scheme for the auxiliary parameters is derived and calibrated using the H-rank algorithm, which provides a concise measure of algebraic complexity in sequences generated by dynamical systems. Numerical simulations reveal hierarchical structures of Arnold tongues of divergence together with characteristic divergence rates of the auxiliary parameters. The hierarchical level of a given auxiliary parameter is identified as a key quantity determining the algebraic complexity of the transient dynamics, with potential implications for information encoding in applications exploiting transient dynamical processes. Full article

This article addresses the trajectory tracking control problem for quadrotor unmanned aerial vehicles (UAVs) subject to complex external disturbances and parameter uncertainties. To balance disturbance rejection with control signal smoothness, a practical predefined-time control scheme incorporating variable exponent coefficients (VEC) is proposed. First, a variable exponent practical predefined-time disturbance observer (VEC-PPTDO) is designed to dynamically estimate and compensate for unknown aerodynamic disturbances. Additionally, a practical predefined-time fractional-order sliding mode control (VEC-PPTFOSMC) scheme is developed, which fuses fractional-order calculus with VEC reaching laws to accelerate convergence and mitigate high-frequency chattering. Based on Lyapunov stability theory, the practical predefined-time stability of the entire closed-loop system is rigorously proven. Finally, comparative simulations under severe stochastic disturbances validate the proposed framework. Quantitative results demonstrate that the proposed scheme achieves a steady-state convergence time of 0.95 s. Compared to the integer-order benchmarks, the proposed method reduces the convergence time by an average of 15.2%, while decreasing the root mean square error (RMSE) and integral absolute error (IAE) by an average of 13.4% and 14.5%, respectively. Consequently, the proposed architecture enhances the dynamic tracking precision, control efficiency, and operational robustness of the quadrotor system. Full article

Liquid biopsy has evolved beyond its original role as a minimally invasive approach for mutation detection and is now being developed as a broader analytical framework for cancer detection, stratification, and longitudinal monitoring. Improvements in next-generation sequencing, assay chemistry, and computational analysis have increased analytical sensitivity, including in settings with low tumor fraction and very low variant allele abundance. These advances have expanded the utility of cfDNA analysis in measurable residual disease assessment and in the detection of low-abundance tumor-derived signals across multiple clinical contexts. At the same time, the field has shifted toward interpreting cfDNA as a carrier of higher-order biological information rather than solely a substrate for mutation calling. Fragmentation profiles, nucleosome positioning, and chromatin accessibility patterns derived from plasma DNA have been used to infer transcriptional and regulatory states, raising the possibility that cfDNA may capture functional tumor states not readily accessible through genotype-focused assays alone. These developments have prompted growing interest in chromatin-informed cfDNA analysis as a means of identifying pathway activity, enhancer usage, transcription factor occupancy, and potentially actionable biological dependencies. However, the translational relevance of many such inferences remains incompletely established, and preanalytical variability, limited cross-cohort generalizability, and the gap between analytical performance and clinical utility continue to constrain clinical translation. This review examines the role of cfDNA in adaptive oncology, highlighting recent analytical advances, assessing the current evidence supporting their biological and clinical utility, and considering the extent to which cfDNA-derived regulatory inference may contribute to adaptive oncology and therapeutic decision-making. Full article

Emerging technologies and cyber–physical systems have led to the development of complex mathematical models described by differential equations with multiple fractional orders. In this regard, this paper investigates the stability of control systems for this class of models, defined by state equations with multiple fractional orders ranging between 0 and 1. Matrix criteria and comparison principle for linear and nonlinear autonomous systems of different fractional orders are developed based on generalized Lyapunov functions for differential equations with multi-order fractional exponents. The results are extended to non-autonomous linear systems or systems with nonlinear components of different fractional orders. Application of the Yakubovich–Kalman–Popov lemma, adapted for this class of systems, allows us to obtain new stability criteria presented as frequency criteria and represented graphically by familiar frequency plots similar those of the Nyquist or Popov type. Numerical applications illustrate these results, such as models of complex human–machine systems described by state equations of multivariable fractional orders. An analysis of the advantages of the proposed methods compared to procedures and techniques used in other papers regarding the study of multi-order fractional exponent systems is presented. It is demonstrated that the proposed methods minimize the computational effort required for stability criteria. Full article

This work extends the analytical operation of the Riemann–RHPHilbert approach (RHA) for fractional-order nonlinear integrable systems under the solvable meaning of inverse scattering transform (IST) to variable-coefficient fractional-order nonlinear models. Firstly, based on the matrix spectral problem proposed by Ablowitz, Kaup, Newell, and Segur, this article derives an integer-order integrable system, which is abbreviated as the AKNS hierarchy. Secondly, by taking specific values of the operator in the derived AKNS hierarchy, a variable-coefficient fractional higher-order NLS hierarchy (vfhNLSH) is obtained, and its anomalous dispersion relation (ADR) is derived via formal solution. Significantly, the reductions of the vfhNLSH include three variable-coefficient fractional-order integrable models: the Hirota equation (vfHE), the Lakshmanan–Porsezian–Daniel equation (vfLPDE), and the fifth-order NLS equation (vffNLSE). Finally, we conduct a detailed study on the representative vfHE as an example rather than a special case and construct its explicit N-fold analytical solution based on the extension of the RHA. At the same time, numerical visualization simulations are conducted to demonstrate the waveform structure characteristics of the solutions under $N=1$ and $N=2$ conditions, including solitons, breathers, and their coupled nonlinear waves. The same process is fully applicable to the other two reduced models, with only some differences in the related results and the dynamic behavior of the solutions. It is shown that the temporal part of the Lax pair associated with the vfHE cannot yet be explicitly determined. Therefore, the fractional-order extension of the RHA presented in this article constitutes a formal or RHA-inspired construction, rather than a fully rigorous fractional-order RHA extension. Full article

This paper is concerned with the numerical solution of the variable-order time fractional advection–diffusion–reaction equation (VO-TFADRE) in two space dimensions. We first propose a Crank–Nicolson (C-N) discretization scheme based on central difference operators and L1 formula for space and time variables, respectively. Then, we apply the C-N scheme to construct a new algorithm, namely the explicit group (EG) method, for the model problem under consideration. The EG method utilizes the idea of small fixed-size groups of mesh points and comes with computational merits as compared with the C-N scheme. Stability and convergence analyses are given in this work. The resulting discretization leads to large sparse linear systems, which are solved using the Bi-CGSTAB iterative method. Numerical experiments demonstrate that both the C–N and EG schemes achieve accurate approximations, while the EG method significantly reduces computational time. To economize further on the computational cost, we propose a parallelized version of the EG method for solving the VO-TFADRE. Carried out numerical simulations reveal that the parallel algorithm is more efficient than the serial algorithm for solving the problem under consideration. Full article

We investigate chaos control and hybrid synchronization in a variable-order fractional Arneodo system by treating the differentiation order $\alpha \left(t\right)$ as a closed-loop control variable. A hybrid chaos indicator, combining a tracking error with a windowed estimate of the largest Lyapunov exponent, drives both static and dynamic order modulation laws. The presence and uniqueness of solutions are demonstrated through two distinct methodologies: a piecewise constant-order decomposition with an explicit convergence rate and a direct contraction-mapping argument on the variable-order Volterra operator. Local stability is analyzed via Matignon’s spectral criterion under a quasi-static (frozen-time) approximation. The modulation laws are designed to steer $\alpha \left(t\right)$ below the critical order ${\alpha }_c\approx 0.8632$, at which the nontrivial equilibria $E_{1,2}=(\pm \sqrt{5.5},0,0)$ become locally asymptotically stable. A second-order predictor–corrector scheme attains its expected convergence rate. A controlled ablation study over 200 Monte Carlo runs demonstrates that the proposed laws reduce the terminal tracking error by 81% relative to the best fixed-order baseline, while requiring approximately eight orders of magnitude less control effort than classical active control. Hybrid synchronization (complete in $(u,v)$ and anti-synchronization in w) is successfully achieved in the variable-order setting. Full article