Search Results (1,018)

We study a nonlinear fractional differential equation, defined on a finite and infinite interval. In the finite interval setting, we attach initial conditions and parameter-dependent boundary conditions to the problem. We apply a dichotomy approach, coupled with the numerical-analytic method, to analyze the problem and to construct a sequence of approximations. Additionally, we study the existence of bounded solutions in the case when the fractional differential equation is defined on the half-axis and is subject to asymptotic conditions. Our theoretical results are applied to the Arctic gyre equation in the fractional setting on a finite interval. Full article

The asymptotic behavior of discrete Riemann–Liouville fractional difference equations is a fundamental problem with important mathematical and physical implications. In this paper, we investigate a particular case of such an equation of the order $0.5$ subject to a given initial condition. We establish the existence of a unique solution expressed via a Mittag-Leffler-type function. The delta-asymptotic behavior of the solution is examined, and its convergence properties are rigorously analyzed. Numerical experiments are conducted to illustrate the qualitative features of the solution. Furthermore, a neural network-based approximation is employed to validate and compare with the analytical results, confirming the accuracy, stability, and sensitivity of the proposed method. Full article

Prefabricated unidirectional carbon fiber reinforced polymer (CFRP) composite strips are extensively used as a means of infrastructure rehabilitation through adhesive bonding to the external surface of structural concrete elements. Most data to date are from laboratory tests ranging from a few months to 1–2 years providing an insufficient dataset for prediction of long-term durability. This investigation focuses on the assessment of the response of three different prefabricated CFRP systems exposed to water, seawater, and alkaline solutions for 5 years of immersion in deionized water conducted at three temperatures of 23, 37.8 and 60 °C, all well below the glass transition temperature levels. Overall response is characterized through tensile and short beam shear (SBS) testing at periodic intervals. It is noted that while the three systems are similar, with the dominant mechanisms of deterioration being related to matrix plasticization followed by fiber–matrix debonding with levels of matrix and interface deterioration being accelerated at elevated temperatures, their baseline characteristics and distributions are different emphasizing the need for greater standardization. While tensile modulus does not degrade appreciably over the 5-year period of exposure with final levels of deterioration being between 7.3 and 11.9%, both tensile strength and SBS strength degrade substantially with increasing levels based on temperature and time of immersion. Levels of tensile strength retention can be as low as 61.8–66.6% when immersed in deionized water at 60 °C, those for SBS strength can be 38.4–48.7% at the same immersion condition for the three FRP systems. Differences due to solution type are wider in the short-term and start approaching asymptotic levels within FRP systems at longer periods of exposure. The very high levels of deterioration in SBS strength indicate the breakdown of the materials at the fiber–matrix bond and interfacial levels. It is shown that the level of deterioration exceeds that presumed through design thresholds set by specific codes/standards and that new safety factors are warranted in addition to expanding the set of characteristics studied to include SBS or similar interface-level tests. Alkali solutions are also shown to have the highest deteriorative effects with deionized water having the least. Simple equations are developed to enable extrapolation of test data to predict long term durability and to develop design thresholds based on expectations of service life with an environmental factor of between 0.56 and 0.69 for a 50-year expected service life. Full article

This paper presents a distributed optimization algorithm for time-varying objective functions utilizing a prescribed-time convergent multi-agent system within undirected communication networks. Departing from conventional time-invariant optimization paradigms with static optimal solutions, our approach specifically addresses the challenge of tracking dynamic optimal trajectories in evolving environments. A novel continuous-time distributed optimization algorithm is developed based on prescribed-time consensus, guaranteeing the consensus attainment among agents within a user-defined timeframe while asymptotically converging to the time-dependent optimal solution. The proposed methodology enables explicit predetermination of convergence duration, representing a significant advancement over existing asymptotic convergence methods. Moreover, two simulation examples on the rendezvous problem and multi-robots control are presented to validate the theoretical results, exhibiting precise time-controlled convergence characteristics and effective tracking performance for time-varying optimization targets. Full article

We investigate a generalized quantum Schrödinger equation in a comb-like structure that imposes geometric constraints on spatial variables. The model is extended by the introduction of nonlocal and fractional potentials to capture memory effects in both space and time. We consider four distinct scenarios: (i) a time-dependent nonlocal potential, (ii) a spatially nonlocal potential, (iii) a combined space–time nonlocal interaction with memory kernels, and (iv) a fractional spatial derivative, which is related to distributions asymptotically governed by power laws and to a position-dependent effective mass. For each scenario, we propose solutions based on the Green’s function for arbitrary initial conditions and analyze the resulting quantum dynamics. Our results reveal distinct spreading regimes, depending on the type of non-locality and the fractional operator applied to the spatial variable. These findings contribute to the broader generalization of comb models and open new questions for exploring quantum dynamics in backbone-like structures. Full article

The authors investigate an epidemic model described by a differential equation, which includes a piecewise constant argument of the generalized type (DEPCAG). In this work, the main goal is to find an invariant region for the system and prove the existence and uniqueness of solutions with the defined conditions using integral equations. On top of that, an auxiliary result is established, outlining the relationship between the unknown function values in the deviation argument and the time parameter. The stability analysis is conducted using the Lyapunov–Razumikhin method, adapted for differential equations with a piecewise constant argument of the generalized type. The trivial equilibrium’s stability is examined, and the stability of the positive equilibrium is assessed by transforming it into a trivial form. Finally, sufficient conditions for the uniform asymptotic stability of both the trivial and positive equilibria are established. Full article

Modern fighter aircraft have an increasing need for at least a moderate level of stealth, and the shape design must bear a part of this constraint. However, the high frequency of close range radar makes high-fidelity radar cross-section computation methods such as the boundary element method too expensive to use in a gradient-free optimization process. On the other hand, asymptotic methods are not able to accurately predict the RCS of complex shapes such as intake cavities. Hence, the need arises for efficient and accurate methods to compute the gradient of high-fidelity radar cross-section computation methods with respect to shape parameters. In this paper, we propose an adjoint formulation for the boundary element method to efficiently compute these gradients. We present a novel approach to calculate the gradient of high-fidelity radar cross-section computations using the boundary element method. Our method employs an adjoint formulation that allows for the efficient computation of these gradients. This is particularly beneficial in shape optimization problems where accurate and efficient methods are crucial to designing modern fighter aircraft with stealth capabilities. By avoiding the need for solving the actual adjoint problem in certain cases, our formulation provides a more streamlined solution while still maintaining high accuracy. We demonstrate the effectiveness of our method by performing shape optimization on various shapes, including simple geometries like spheres and ellipsoids, as well as complex aircraft shapes with multiple variables. Full article

This work will present an algorithmic approach for robust control focusing on hydraulic–mechanical systems. The approach is applied to a hydraulic actuator driving a cart with an inverted pendulum. The algorithmic approach aims to satisfy two robust control requirements for single input single output (SISO) linear systems with nonlinear uncertain structure. The first control requirement is robust stabilization, and the second is robust asymptotic command following for arbitrary reference signals. The approach is analyzed in two stages. In the first stage, the stability regions of the controller parameters are identified. In the second stage, a Particle Swarm Optimization Algorithm (PSO) is applied to find suboptimal solutions for the controller parameters in these regions, with respect to a suitable performance cost function. The application of the approach to a hydraulic actuator, driving a cart with an inverted pendulum, satisfies the goal of achieving precise control of the pendulum angle, despite the system’s inherent physical uncertainties. Full article

A two-dimensional time-dependent model is presented for upward-propagating internal gravity waves generated by an imposed thermal forcing in a layer of fluid with uniform background velocity and stable stratification under the anelastic approximation. The configuration studied is representative of a situation with deep or shallow latent heating in the lower atmosphere where the amplitude of the waves is small enough to allow linearization of the model equations. Approximate asymptotic time-dependent solutions, valid for late time, are obtained for the linearized equations in the form of an infinite series of terms involving Bessel functions. The asymptotic solution approaches a steady-amplitude state in the limit of infinite time. A weakly nonlinear analysis gives a description of the temporal evolution of the zonal mean flow velocity and temperature resulting from nonlinear interaction with the waves. The linear solutions show that there is a vertical variation of the wave amplitude which depends on the relative depth of the heating to the scale height of the atmosphere. This means that, from a weakly nonlinear perspective, there is a non-zero divergence of vertical momentum flux, and hence, a non-zero drag force, even in the absence of vertical shear in the background flow. Full article

A mathematical model of the anomalous diffusion of surfactant and the process of adsorption–desorption on an interface is analyzed using a fractional calculus approach. The model is based on time-fractional partial differential equations in the bulk phases and the corresponding time-fractional description of the flux bulk–interface. The general case, when the surfactant is soluble in both phases, is considered under the assumption that the adsorption–desorption process is diffusion-controlled. Some of the most popular kinetic models of Henry, Langmuir, and Volmer are considered. Applying the Laplace transform, the partial differential model is transformed into a single multi-term time-fractional nonlinear ordinary differential equation for the surfactant concentration on the interface. Based on existing analytical solutions of linear time-fractional differential equations, the exact solution in the case of the Henry model is derived in terms of multinomial Mittag–Leffler functions, and its asymptotic behavior is studied. Further, the fractional differential model in the general nonlinear case is rewritten as an integral equation, which is a generalization of the well-known Ward–Tordai equation. For computer simulations, based on the obtained integral equation, a predictor–corrector numerical technique is developed. Numerical results are presented and analyzed. Full article

We consider a compression problem in a multi-source status-updating system through a representative two-source scenario. The status updates are generated by two independent sources following heterogeneous Poisson processes. These updates are then compressed into binary strings and sent to the receiver via a shared, error-free channel with a unit rate. We propose two compression schemes—a multi-quantizer compression scheme, where a dedicated quantizer–encoder pair is assigned to each source for compression, and a single-quantizer compression scheme, employing a unified quantizer–encoder pair shared across both sources. For each scheme, we formulate an optimization problem to jointly design quantizer–encoder pairs, with the objective of minimizing the sum of the average ages subject to a distortion constraint of symbols, respectively. The following three theoretical results are established: (1) The combination of two uniform quantizers with different parameters, along with their corresponding AoI-optimal encoders, provides an asymptotically optimal solution for the multi-quantizer compression scheme. (2) The combination of a piecewise uniform w-quantizer with an AoI-optimal encoder provides an asymptotically optimal solution for the single-quantizer compression scheme. (3) For both schemes, the optimal sum of the average ages is asymptotically linear with respect to the log distortion, with the same slope determined by the sources’ arrival rates. Full article

The behavior of the Rucklidge-type dynamical system was investigated, providing some semi-analytical solutions, in this paper. This system was analytically investigated by means of the Optimal Auxiliary Functions Method (OAFM) for two cases. An exact parametric solution was obtained. The effect of the physical parameters was investigated on the asymptotic behaviors and damped oscillations of the solutions. Damped oscillations are essential for analyzing and designing various mechanical, biological, and electrical systems. Many of the applications involving these systems represent the main reason of this work. A comparison between the obtained results via the OAFM, the analytical solution obtained with the iterative method, and the corresponding numerical solution was performed. The accuracy of the analytical and corresponding numerical results is illustrated by graphical and tabular representations. Full article

Conventional integer-order models fail to adequately capture non-local memory effects and constrained nonlinear interactions in emotional dynamics. To address these limitations, we propose a coupled framework that integrates Caputo fractional derivatives with hyperbolic tangent–based interaction functions. The fractional-order term quantifies power-law memory decay in affective states, while the nonlinear component regulates connection strength through emotional difference thresholds. Mathematical analysis establishes the existence and uniqueness of solutions with continuous dependence on initial conditions and proves the local asymptotic stability of network equilibria ($W_{ij}^{*}={\displaystyle \frac{1}{\delta }}{\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )$, e.g., $W^{*}\approx 1.40$ under typical parameters $\eta =0.5$, $\delta =0.3$). We further derive closed-form expressions for the steady-state variance under stochastic perturbations ($\mathrm{Var}\left(W_{ij}\right)={\displaystyle \frac{{\sigma }_{\zeta }^2}{2\eta \delta }}$) and demonstrate a less than 6% deviation between simulated and theoretical values when ${\sigma }_{\zeta }=0.1$. Numerical experiments using the Euler–Maruyama method validate the convergence of connection weights toward the predicted equilibrium, reveal Gaussian features in the stationary distributions, and confirm power-law scaling between noise intensity and variance. The numerical accuracy of the fractional system is further verified through L1 discretization, with observed error convergence consistent with theoretical expectations for $\mu =0.5$. This framework advances the mechanistic understanding of co-evolutionary dynamics in emotion-modulated social networks, supporting applications in clinical intervention design, collective sentiment modeling, and psychophysiological coupling research. Full article

In the continuous drilling process of oil wells, to achieve the efficient screening of drilling fluids by the vibrating screen while ensuring the safety of the screening operation, an anti-resonance system driven by two exciters with triple-frequency (denoted as 3:1 frequency ratio) is proposed. Initially, differential motion equations are formulated utilizing Lagrange’s equation, followed by the definition of vibration isolation coefficients adopting ratios. Triple-frequency synchronization and stability criterion between two eccentric blocks are subsequently elucidated via the asymptotic method and Routh–Hurwitz criterion. Concurrently, the effects of structural parameters on vibration isolation capacity, steady-state trajectory, and the triple-frequency synchronization phase are investigated through numerical computation. Ultimately, the reliability of the theoretical study is corroborated by simulation analysis. Results indicate that under the allowable system parameters for the practical project, the amplitude of the vibration body can exceed three times that of the isolation body; the two solutions of the stable phase difference (SPD) are different by π, one of which is stable and the other is unstable, and the stability of phase difference is determined by the sign of the stability coefficient. This work is useful for developing new vibrating screens and other multi-frequency vibration machines. Full article

Aimed at dealing with the problems of high reliability solution cost and low solution accuracy under random excitation, especially Gaussian white noise excitation, this paper proposes a probability density evolution and reliability analysis method for nonlinear gear transmission systems under Gaussian white noise excitation based on the path integration method. This method constructs an efficient probability density evolution framework by combining the path integration method, the Chapman–Kolmogorov equation, and the Laplace asymptotic expansion method. Based on Rice’s theory and combined with the adaptive Gauss–Legendre integration method, the transient and cumulative reliability of the system are path integration method calculated. The research results show that in the periodic response state, Gaussian white noise leads to the diffusion of probability density and peak attenuation, and the system reliability presents a two-stage attenuation characteristic. In the chaotic response state, the intrinsic dynamic instability of the system dominates the evolution of the probability density, and the reliability decreases more sharply. Verified by Monte Carlo simulation, the method proposed in this paper significantly outperforms the traditional methods in both computational efficiency and accuracy. The research reveals the coupling effect of Gaussian white noise random excitation and nonlinear dynamics, clarifies the differences in failure mechanisms of gear systems in periodic and chaotic states, and provides a theoretical basis for the dynamic reliability design and life prediction of nonlinear gear transmission systems. Full article