Search Results (21)

This work presents a comprehensive mathematical framework for symmetrized neural network operators operating under the paradigm of fractional calculus. By introducing a perturbed hyperbolic tangent activation, we construct a family of localized, symmetric, and positive kernel-like densities, which form the analytical backbone for three classes of multivariate operators: quasi-interpolation, Kantorovich-type, and quadrature-type. A central theoretical contribution is the derivation of the Voronovskaya–Santos–Sales Theorem, which extends classical asymptotic expansions to the fractional domain, providing rigorous error bounds and normalized remainder terms governed by Caputo derivatives. The operators exhibit key properties such as partition of unity, exponential decay, and scaling invariance, which are essential for stable and accurate approximations in high-dimensional settings and systems governed by nonlocal dynamics. The theoretical framework is thoroughly validated through applications in signal processing and fractional fluid dynamics, including the formulation of nonlocal viscous models and fractional Navier–Stokes equations with memory effects. Numerical experiments demonstrate a relative error reduction of up to 92.5% when compared to classical quasi-interpolation operators, with observed convergence rates reaching $\mathcal{O}\left(n^{-1.5}\right)$ under Caputo derivatives, using parameters $\lambda =3.5$, $q=1.8$, and $n=100$. This synergy between neural operator theory, asymptotic analysis, and fractional calculus not only advances the theoretical landscape of function approximation but also provides practical computational tools for addressing complex physical systems characterized by long-range interactions and anomalous diffusion. Full article

Magneto-electro-elastic materials, a novel class of smart materials, exhibit remarkable energy conversion properties, making them highly suitable for applications in nanotechnology. This study focuses on various aspects of the fractional nonlinear longitudinal wave equation (FNLWE) that models wave propagation in a magneto-electro-elastic circular rod. Using the direct algebraic method, several new soliton solutions were derived under specific parameter constraints. In addition, Galilean transformation was employed to explore the system’s sensitivity and quasi-periodic dynamics. The study incorporates 2D, 3D, and time-series visualizations as effective tools for analyzing quasi-periodic behavior. The results contribute to a deeper understanding of the nonlinear dynamical features of such systems and demonstrate the robustness of the applied methodologies. This research not only extends existing knowledge of nonlinear wave equations but also introduces a substantial number of new solutions with broad applicability. Full article

This study focuses on optimizing the thermal performance of hydrogen turbine blades through a sensitivity analysis using generalized fractional calculus. The approach is designed to capture the transient temperature dynamics and optimize thermal profiles by analyzing the influence of a fractional-order parameter on the system’s behavior. The model was implemented in Python, using Monte Carlo simulations to evaluate the impact of the parameter on the temperature evolution in different thermal regimes. Three distinct regions were identified: the Quasi-Uniform Region (where fractional effects are negligible), the Sub-Classical Region (characterized by delayed thermal behavior), and the Super-Classical Region (exhibiting enhanced heat accumulation). Regression analyses reveal quadratic and cubic dependencies of blade temperature on the fractional-order parameter, confirming the robustness of the model with R² values greater than 0.96. The study highlights the potential of using fractional calculus to optimize the thermal response of turbine blades, helping to identify the most suitable parameters for faster stabilization and efficient heat management in hydrogen turbines. Furthermore, it was found that by adjusting the fractional-order parameter, the system can be optimized to reach equilibrium more rapidly while achieving higher temperatures. Importantly, the equilibrium is not altered but rather accelerated based on the chosen parameter, ensuring a more efficient thermal stabilization process. Full article

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (${\mathcal{l}}^{\mathtt{q}(·)}\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$). Moreover, it was developed using classical Lebesgue space (${\mathtt{L}}_{\mathtt{p}}$) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not. Full article

This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area of solid continuity that sculpts the compactness area. The conformable Hilfer fractional time derivative, the theory of fuzzy sets, and the infinite delay arguments support the solution set’s controllability. We explain the existence due to the convergence properties of Mittage–Leffler functions (${\mathbb{E}}_{\alpha ,\beta }$), that is, hatching the existing arguments according to FMQHI and the continuity of infinite delay, which has not been presented before. To prove the main results, we apply the Leray–Schauder nonlinear alternative thereom in the interpolation of Banach spaces. This problem seems to draw new extents on the controllability field of stochastic dynamic models. Full article

Magnetic suspended dual-rotor systems (MSDS) provide the potential to significantly improve the performance of aero-engines by eliminating the wear and lubrication system, and solve vibration control issues effectively. However, the nonlinear dynamics of MSDS with rubbing is rarely investigated. In this work, the nonlinear support characteristics of active magnetic bearings (AMBs) are described by the equivalent magnetic circuit method, the impact force is characterized by the Lankarani–Nikravesh model, and the nonlinear dynamic model is established using the finite element method. On this basis, the influence of speed ratio on the nonlinear dynamics is investigated. Simulation results show that the fundamental sub-synchronous vibration of period n is the dominant motion of MSDS, where n is determined by the speed ratio. The frequency components of sub-synchronous vibrations of period k are integer multiples of the minimum dimensionless frequency component 1/k, where k is a positive integral multiple of n. Quasi-periodic and chaotic vibrations are more likely to occur near critical speeds, and their main frequency components can be expressed as a variety of combined frequency components of the rotating frequency difference and its fractional frequency. To reduce the severity of fluctuating stresses stemming from complicated non-synchronous vibrations, speed ratios, corresponding to smaller n and AMB control parameters attenuating vibration amplitude or avoiding critical speeds, are suggested. Full article

As a typical brittle material, the tensile strength of concrete is much lower than its compressive strength. The main failure mode of concrete buildings under explosive and impact loading is spalling, so it is crucial to understand the dynamic tensile performance of concrete. This paper presents an experimental study on the dynamic tensile strength of steel-fiber-reinforced self-compacting concrete (SFRSCC). Specimens of two different self-compacting concrete (SCC) mixes (C40 and C60) and four different fiber volume fractions (0.5%, 1.0%, 1.5%, and 2.0%) are fabricated. Dynamic tensile strengths of SFRSCC are obtained using a modified Hopkinson bar system. The relationships between the dynamic tensile strength of the corresponding SCC mix, the quasi-static compressive strength, and the fiber volume fraction are discussed. An empirical equation is proposed. It is shown that SFRSCC with high compressive strength has higher dynamic tensile strength than low-strength SFRSCC for the same fiber content, and the dynamic tensile strength of SFRSCC possesses an approximately linear relation with the fiber volume fraction. The mechanism underlying this fiber-reinforcement effect is investigated. Full article

Considering the difficulty of receiving small signals under strong electromagnetic jamming, this paper proposes a small-signal anti-jamming scheme based on a single dynamic metamaterial antenna (DMA). Our scheme uses the dynamic-adjustable characteristics of the DMA to perform spatial filtering at the antenna radio frequency (RF) front-end, to suppress strong jamming signals in advance and to improve the receiver’s ability to receive and demodulate small signals. Specifically, we take the maximization of signal-to-interference-plus-noise ratio (SINR) as the optimization goal, transform the fractional non-convex objective function model into a quasi-convex semi-definite relaxation (SDR) problem, and use the Charnes-Cooper (CC) transform algorithm to find the optimal DMA array-element codeword-state matrix. Simulation results show that DMA has better spatial-beamforming capability than traditional antenna arrays, and the proposed scheme can better resist strong jamming. DMA realizes the effect of digital beamforming at the back end of the traditional communication system, has the advantages of traditional digital-spatial filtering, and further improves the receiver’s ability to receive and demodulate small signals. Full article

This study investigates the multistability phenomenon and coexisting attractors in the modified Autonomous Van der Pol-Duffing (MAVPD) system and its fractional-order form. The analytical conditions for existence of periodic solutions in the integer-order system via Hopf bifurcation are discussed. In addition, conditions for approximating the solutions of the fractional version to periodic solutions are obtained via the Hopf bifurcation theory in fractional-order systems. Moreover, the technique for hidden attractors localization in the integer-order MAVPD is provided. Therefore, motivated by the previous discussion, the appearances of self-excited and hidden attractors are explained in the integer- and fractional-order MAVPD systems. Phase transition of quasi-periodic hidden attractors between the integer- and fractional-order MAVPD systems is observed. Throughout this study, the existence of complex dynamics is also justified using some effective numerical measures such as Lyapunov exponents, bifurcation diagrams and basin sets of attraction. Full article

We study an air-fluidized granular monolayer composed of plastic spheres which roll on a metallic grid. The air current is adjusted so that the spheres never lose contact with the grid and so that the dynamics may be regarded as pseudo two dimensional (or two dimensional, if the effects of the sphere rolling are not taken into account). We find two surprising continuous transitions, both of them displaying two coexisting phases. Moreover, in all the cases, we found the coexisting phases display a strong energy non-equipartition. In the first transition, at a weak fluidization, a glass phase coexists with a disordered fluid-like phase. In the second transition, a hexagonal crystal coexists with the fluid phase. We analyze, for these two-phase systems, the specific diffusive properties of each phase, as well as the velocity correlations. Surprisingly, we find a glass phase at a very low packing fraction and for a wide range of granular temperatures. Both phases are also characterized by strong anticorrelated velocities upon a collision. Thus, the dynamics observed for this quasi two-dimensional system unveil phase transitions with peculiar properties, very different from the predicted behavior in well-know theories for their equilibrium counterparts. Full article

In this paper, a battery pack and a supercapacitor bank hybrid energy storage system (HESS) with a new control configuration is proposed for electric vehicles (EVs). A bidirectional quasi-Z-source inverter (Bq-ZSI) and a bidirectional DC-DC converter are used in the powertrain of the EV. The scheme of the control for the proposed HESS Bq-ZSI using finite control set model predictive control (FCS-MPC) is first deduced to enhance the dynamic performance. With the idea of managing battery degradation mitigation, the fractional-order PI (FOPI) controller is then applied and associated with a filtering technique. The Opal-RT-based real-time simulation is next executed to verify the performance and effectiveness of the proposed HESS control strategy. As a result, the proposed HESS Bq-ZSI with this control scheme provides a quick response to the mechanical load and stable DC link voltage under the studied driving cycle. Moreover, the comparative results also show that the proposed HESS Bq-ZSI equipped with the new control configuration enables the reduction of the root-mean-square value, the mean value, and the standard deviation by 57%, 59%, and 27%, respectively, of the battery current compared to the battery-based inverter. Thus, the proposed HESS Bq-ZSI using these types of controllers can help to improve the EV system performance. Full article

This is a review addressing soliton-like states in systems with nonlocal nonlinearity. The work on this topic has long history in optics and related areas. Some results produced by the work (such as solitons supported by thermal nonlinearity in optical glasses, and orientational nonlinearity, which affects light propagation in liquid crystals) are well known, and have been properly reviewed in the literature, therefore the respective models are outlined in the present review in a brief form. Some other studies, such as those addressing models with fractional diffraction, which is represented by a linear nonlocal operator, have started more recently, therefore it will be relevant to review them in detail when more results will be accumulated; for this reason, the present article provides a short outline of the latter topic. The main part of the article is a summary of results obtained for two-dimensional solitons in specific nonlocal nonlinear models originating in studies of Bose–Einstein condensates (BECs), which are sufficiently mature but have not yet been reviewed previously (some results for three-dimensional solitons are briefly mentioned too). These are, in particular, anisotropic quasi-2D solitons supported by long-range dipole-dipole interactions in a condensate of magnetic atoms and giant vortex solitons (which are stable for high values of the winding number), as well as 2D vortex solitons of the latter type moving with self-acceleration. The vortex solitons are states of a hybrid type, which include matter-wave and electromagnetic-wave components. They are supported, in a binary BEC composed of two different atomic states, by the resonant interaction of the two-component matter waves with a microwave field that couples the two atomic states. The shape, stability, and dynamics of the solitons in such systems are strongly affected by their symmetry. Some other topics are included in the review in a brief form. This review uses the “Harvard style” of referring to the bibliography. Full article

This research explores the solitary wave solutions, including dynamic transitions for a fractional low-pass electrical transmission (LPET) line model. The fractional-order (FO) LPET line mathematical system has yet to be published, and neither has it been addressed via the extended direct algebraic technique. A computer program is utilized to validate all of the incoming solutions. To illustrate the dynamical pattern of a few obtained solutions indicating trigonometric, merged hyperbolic, but also rational soliton solutions, dark soliton solutions, the representatives of the semi-bright soliton solutions, dark singular, singular solitons of Type 1 and 2, and their 2D and 3D trajectories are presented by choosing appropriate values of the solutions’ unrestricted parameters. The effects of fractionality and unrestricted parameters on the dynamical performance of achieved soliton solutions are depicted visually and thoroughly explored. We furthermore discuss the sensitivity assessment. We, however, still examine how our model’s perturbed dynamical framework exhibits quasi periodic-chaotic characteristics. Our investigated solutions are compared with those listed in published literature. This research demonstrates the approach’s profitability and effectiveness in extracting a range of wave solutions to nonlinear evolution problems in mathematics, technology, and science. Full article

This study aims to identify soliton structures as an inherent fractional discrete nonlinear electrical transmission lattice. Here, the analysis is founded on the idea that the electrical properties of a capacitor typically contain a non-integer-order time derivative in a realistic system. We construct a non-integer order nonlinear partial differential equation of such voltage dynamics using Kirchhoff’s principles for the model under study. It was discovered that the behavior for newly generated soliton solutions is impacted by both the non-integer-order time derivative and connected parameters. Regardless of structure, the fractional-order alters the propagation velocity of such a voltage wave, thus bringing up a localized framework under low coupling coefficient values. The generalized auxiliary equation method drove us to these solitary structures while employing the modified Riemann–Liouville derivatives and the non-integer order complex transform. As well as addressing sensitivity testing, we also investigate how our model’s altered dynamical framework shows quasi-periodic properties. Some randomly selected solutions are shown graphically for physical interpretation, and conclusions are held at the end. Full article

Owing to the symmetry between drive–response systems, the discussions of synchronization performance are greatly significant while exploring the dynamics of neural network systems. This paper investigates the quasi-synchronization (QS) and quasi-uniform synchronization (QUS) issues between the drive–response systems on fractional-order variable-parameter neural networks (VPNNs) including probabilistic time-varying delays. The effects of system parameters, probability distributions and the order on QS and QUS are considered. By applying the Lyapunov–Krasovskii functional approach, Hölder’s inequality and Jensen’s inequality, the synchronization criteria of fractional-order VPNNs under controller designs with constant gain coefficients and time-varying gain coefficients are derived. The obtained criteria are related to the probability distributions and the order of the Caputo derivative, which can greatly avoid the situation in which the upper bound of an interval with time delay is too large yet the probability of occurrence is very small, and information such as the size of time delay and probability of occurrence is fully considered. Finally, two examples are presented to further confirm the effectiveness of the algebraic criteria under different probability distributions. Full article