Search Results (1,155)

This paper introduces a rigorous class of two-dimensional Markov-switching autoregressive moving-average (2D MS-ARMA) models for spatial lattice data exhibiting regime-dependent dynamics. The switching mechanism is governed by a latent causal Markov random field that drives spatial transitions between regime-specific autoregressive and moving-average structures. We provide sufficient conditions for the existence of a strictly stationary solution through the top Lyapunov exponent associated with a sequence of random matrices obtained from a state-space representation constructed along the lexicographic order. For the first-order bidirectional specification, we derive explicit spectral conditions linking stationarity to the regime-dependent spectral radii. Sufficient conditions ensuring the existence of finite second-order moments are also provided. Parameter estimation is carried out using a variational expectation–maximization (VEM) algorithm based on a mean-field approximation of the posterior distribution of the hidden regimes. The E-step yields closed-form coordinate ascent updates, while the M-step relies on gradient-based numerical optimization with derivatives computed via recursive differentiation. Under increasing-domain asymptotics, we discuss the consistency and asymptotic behavior of the variational estimator. The proposed framework fills a methodological gap between classical one-dimensional Markov-switching ARMA models and spatial autoregressive structures by extending regime-switching theory to multi-indexed processes with rigorous probabilistic foundations. It provides a comprehensive basis for statistical inference, model diagnostics, and prediction in spatially heterogeneous environments. Full article

As a core load-bearing component, the aero-engine rear cooling plate requires its design to simultaneously meet strength requirements and lightweight indicators. The topology optimization method considering stress constraints is the core technical path to achieve this goal, but it suffers from insufficient control precision in key areas, easily leading to material redundancy. To address this issue, a partitioned topology optimization method based on the multi-feature K-means algorithm is proposed. First, by integrating multi-dimensional features including element stress, physical density, and spatial position, an innovative multi-feature K-means algorithm is employed to realize dynamic adaptive partitioning during the optimization process. Secondly, combined with the p-norm method for partitioned stress aggregation, a precise prediction and control method for partitioned stress is adopted to refine stress constraints. Thirdly, a topology optimization model of the rear cooling plate with multi-feature partitioned stress constraints is constructed, and the adjoint method is used to solve the stress sensitivities under centrifugal loads. Finally, the effectiveness of the proposed method is verified using the rear cooling plate model. The rear cooling plate is discretized with 0.5 mm 2D axisymmetric finite elements, the filter radius is 4 mm, and the Method of Moving Asymptotes (MMA) is employed for the solution. The mass fraction of the finally optimized rear cooling plate structure is 0.157, which is 13.7% lower than that obtained by the global stress constraint method and 11.3% lower than that obtained by the topology optimization method considering both the geometric partitioned stress constraints and global stress constraints. The proposed method provides a new approach for the lightweight design of the aero-engine rear cooling plate. Full article

This paper presents a unified analytical and computational framework for the study of typhoid fever transmission dynamics governed by a Caputo fractional-order compartmental model of order $\kappa \in (0,1]$. The population is stratified into five epidemiological classes, namely susceptible $\left(\mathcal{S}\right)$, asymptomatic $\left(\mathcal{A}\right)$, symptomatic $\left(\mathcal{I}\right)$, hospitalised $\left(\mathcal{H}\right)$, and recovered $\left(\mathcal{R}\right)$, and the governing system explicitly incorporates asymptomatic transmission, treatment dynamics, and temporary immunity with waning. The use of the Caputo fractional derivative is motivated by the well-documented existence of chronic asymptomatic Salmonella Typhi carriers, whose heavy-tailed sojourn times in the carrier state are naturally encoded by the Mittag–Leffler waiting-time distribution arising from the fractional operator. A complete qualitative analysis of the fractional system is carried out: the basic reproduction number ${\mathcal{R}}_0$ is derived via the next-generation matrix method; local and global asymptotic stability of both the disease-free equilibrium ${\mathcal{E}}_0$ (when ${\mathcal{R}}_0\le 1$) and the endemic equilibrium ${\mathcal{E}}^{*}$ (when ${\mathcal{R}}_0>1$) are established using fractional Lyapunov theory and the LaSalle invariance principle; and the normalised sensitivity indices of ${\mathcal{R}}_0$ are computed to identify transmission-amplifying and transmission-suppressing parameters. Existence, uniqueness, and Ulam–Hyers stability of solutions are established via Banach and Leray–Schauder fixed-point arguments. To complement the analytical results, a fractional physics-informed neural network ($\mathcal{PINN}$) framework is developed to simultaneously reconstruct compartmental trajectories and identify unknown biological parameters from sparse synthetic observations. $\mathcal{PINN}$ embeds the L1-Caputo discretisation directly into the training residuals and employs a four-stage Adam–L-BFGS optimisation strategy to recover five trainable parameters $\Theta$ = $\{\varphi ,\mu ,\sigma ,\psi ,\beta \}$ across three fractional orders $\kappa \in \{1.0,0.95,0.9\}$. The estimated parameters show strong agreement with the true values at the classical limit $\kappa =1.0$ ($\mathrm{MAPE}=2.27\%$), with the natural mortality rate $\mu$ recovered with $\mathrm{APE}\le 0.51\%$ and the transmission rate $\beta$ with $\mathrm{APE}\le 3.63\%$ across all fractional orders, confirming the structural identifiability of the model. Pairwise correlation analysis of the learned parameters establishes the absence of equifinality, validating that $\beta$ can be reliably included in the trainable set. Noise robustness experiments under Gaussian perturbations of $1\%$, $3\%$, and $5\%$ demonstrate graceful degradation ($\mathrm{MAPE}$: $0.82\%\to 3.10\%\to 7.31\%$), confirming the reliability of the proposed framework under realistic observational conditions. Full article

In the ElastoHydrodynamic Lubrication (EHL) meshing contact model for rough interfaces with convex–concave textured micro-asperities, the geometric morphology of the meshing interface exhibits pronounced multiscale characteristics: the macroscale manifests as the correlation between Interface-Enriched Lubrication (IEL) performance and meshing Anti-Scuffing Load-Bearing Capacity (ASLBC), while the microscale corresponds to the textured morphology of rough interfaces. In numerical simulations of EHL meshing contact, such cross-scale disparities necessitate solving large-scale systems of analytical solution equations. Assuming periodicity or quasi-periodicity at the microscale, various established methods enable decoupling the macroscopic and microscopic scales, such formalized approaches constitute homogenization theory. However, classical asymptotic assumptions may introduce considerable approximation errors. This study proposes a micro-texture-informed homogenized contact model based on multiscale characterization that incorporates the coupled effects of gear interface meshing forces and thermo-elastic deformations, effectively extending the applicability of classical asymptotic homogenization methods. Full article

This paper develops an extended Cucker–Smale model that integrates nonlinear velocity alignment with state-dependent binding forces to achieve stable, collision-free flocking in multi-agent systems. Our framework introduces two dedicated control mechanisms: a velocity-dissipative term $\left(K_1\right)$ for accelerated convergence, and a distance-regulating term $\left(K_2\right)$ for formation cohesion and collision avoidance, which collectively ensure stable flocking. Rigorous Lyapunov analysis establishes provable guarantees for asymptotic velocity alignment and collision safety under verifiable initial energy conditions. Numerical simulations validate the theoretical predictions for a 20-agent swarm; scalability analysis demonstrates effective coordination in systems of up to 100 agents and reveals that velocity synchronization improves substantially—with errors decreasing by nearly two orders of magnitude—as $K_2$ increases from 0.05 to 0.50. A Pareto-optimal parameter region ($K_2\in [0.15,0.30]$) is identified, which achieves sub-centimeter-per-second alignment accuracy while maintaining energy consumption below 35% of the baseline. The proposed framework provides a theoretically rigorous yet practically viable solution for applications demanding guaranteed safety and precise coordination, such as UAV formations, robotic swarms, and autonomous vehicle platoons. Full article

Asymptotic solutions are derived to model the development of atmospheric internal gravity waves generated by latent heating in a two-dimensional configuration involving a vertically-sheared background flow. The mathematical model comprises nonlinear partial differential equations derived from the conservation laws of fluid dynamics under the anelastic approximation where the background density and temperature vary with altitude. The latent heating is represented by a horizontally-periodic but vertically-localized nonhomogeneous forcing term in the energy conservation equation. This generates gravity waves that are considered as perturbations to the background flow and are expressed as perturbation series, with the leading-order contributions being the solutions of linearized equations. Taking into account the nonlinear terms at the next order gives expressions for the effects of the waves on the background mean flow. Due to the vertical shear, there is a critical level where momentum and energy are transferred from the wave modes to the mean flow. The asymptotic solutions show that the wave–mean-flow interaction is nonlocal and occurs over the range of altitudes from the thermal forcing level up the critical level. This is in contrast to what occurs in the case of waves forced by an oscillatory lower boundary, where the interaction is typically localized around the critical level. It is found that the wave drag is negative above the thermal forcing level, making the mean flow velocity more negative, but it becomes positive as the waves approach the critical level, indicating wave absorption in this region. There is wave transmission through the critical level, as well as absorption, and the extent of transmission depends on the depth of the latent heating profile. The mean potential temperature is reduced above the thermal forcing level and enhanced at the critical level, a situation that could ultimately lead to the development of convective instabilities. Full article

In this paper, we present a mathematical analysis of within-host CHIKV dynamics by developing and studying a novel delay differential equation model that incorporates persistent infected monocytes, discrete time delays, and an antibody-mediated humoral immune response. The model includes five compartments: susceptible monocytes, persistent infected monocytes, actively infected monocytes, CHIKV pathogens, and neutralizing antibodies. To reflect key biological latencies, we introduce four distinct discrete delays accounting for the periods between viral entry and the emergence of infected cell populations, intracellular virion production, and antibody activation. We analyze the model, establishing the positivity, boundedness, and invariance of solutions, and derive the basic reproduction number ${\mathcal{R}}_0$ via the next-generation matrix method. Using Lyapunov functions and LaSalle’s Invariance Principle, we prove a threshold dynamic: the infection-free equilibrium is globally asymptotically stable (GAS) when ${\mathcal{R}}_0\le 1$, while a unique endemic equilibrium is GAS when ${\mathcal{R}}_0>1$. Numerical simulations validate the analytical results and illustrate threshold behavior. A detailed local sensitivity analysis of ${\mathcal{R}}_0$ identifies the most influential parameters, offering theoretical insights into potential intervention strategies. We further investigate the effects of antiviral therapy as a theoretical intervention, deriving a treatment-dependent reproduction number and the critical drug efficacy required for eradication, and explore how the intracellular production delay can itself serve as a critical threshold for infection clearance. The study provides a rigorous theoretical framework that highlights the roles of latency, immune response, and biological delays in CHIKV pathogenesis and offers qualitative insights that may inform future experimental and treatment design studies. Full article

Classical eddy viscosity models add a viscosity term with a turbulent viscosity coefficient developed beginning with the Kolmogorov–Prandtl parameterization. Approximations of unknown accuracy of the unknown mixing lengths and turbulent kinetic energy are typically constructed by solving associated systems of nonlinear convection–diffusion-reaction equations with nonlinear boundary conditions. These often over-diffuse, so additional fixes are added such as wall laws, or different approximations are used in different regions (which must also be specified). Alternately, one can solve an ensemble of NSEs with perturbed data, compute the ensemble mean and fluctuation, and simply directly compute the turbulent viscosity parameterization. This idea is recent. From previous work it seems to be of a lower complexity and greater accuracy. It also produces parameterizations with the correct near-wall asymptotic behavior. The question then arises: Does this ensemble eddy viscosity approach over-diffuse solutions? This question is addressed herein. Full article

This paper presents a systematic asymptotic analysis of the least squares estimator (LSE) for the drift parameter in a fractional stochastic heat equation driven by fractional Brownian motion. Fractional Brownian motion, capable of capturing stylized features in financial markets such as long memory, has become an important modeling tool in financial econometrics and risk management. Based on continuous-time observations of the Fourier coefficients of the solution, we first establish the strong consistency and asymptotic normality of the estimator. We then construct an alternative estimator based on the LSE and analyze its asymptotic behavior. This study provides new asymptotic inference tools for stochastic systems with long-memory properties and extends the theoretical framework for parameter estimation in fractional stochastic partial differential equations. Full article

This paper develops a rigorous inferential framework for a class of Gaussian stochastic processes driven by white noise with constant drift, whose temporal evolution is governed by a Caputo fractional derivative of order ${\displaystyle \alpha \in (1/2,1)}$. The model belongs to the family of fractional Volterra processes, where memory is generated by the dynamics themselves rather than by correlated noise. We derive explicit analytical expressions for the mean, variance, and covariance structure of the solution, thereby characterizing in a precise manner how the fractional order ${\displaystyle \alpha }$ governs both variance growth and the strength of temporal dependence. In particular, the process exhibits correlated increments and a power-law variance scaling of order ${\displaystyle t^{2\alpha -1}}$, highlighting the dual role of ${\displaystyle \alpha }$ as a regularity and memory parameter. Building on this structural analysis, we address the statistical problem of estimating the parameter vector ${\displaystyle (\mu ,\sigma ,\alpha )}$ from discrete-time observations. Two complementary procedures are proposed for the estimation of the fractional order: a variance-growth method based on log–log regression of empirical variances, and a wavelet-based estimator exploiting multi-scale scaling properties of the process. For the drift and diffusion parameters ${\displaystyle (\mu ,\sigma )}$, we construct explicit Gaussian pseudo-maximum likelihood estimators derived from the Volterra covariance structure of the increment process. We establish unbiasedness, ${\displaystyle L^2}$-convergence, strong consistency, and asymptotic normality for all estimators. Furthermore, we derive Berry–Esseen type bounds that quantify the rate of convergence toward the Gaussian law, providing sharp distributional approximations in a genuinely fractional and non-Markovian setting. A Monte Carlo study is carried out, using high-resolution Volterra discretizations, large-scale simulation budgets, covariance-structured linear algebra, and multi-scale diagnostic tools. The numerical experiments confirm the theoretical convergence rates, demonstrate the finite-sample reliability of the estimators, and illustrate the sensitivity of the process dynamics to the fractional order ${\displaystyle \alpha }$: smaller values of ${\displaystyle \alpha }$ produce stronger memory effects and higher variability, while values closer to one lead to smoother and more stable trajectories. The proposed methodology unifies statistical inference for long-memory Gaussian processes with fractional differential stochastic dynamics, offering a coherent analytical and computational framework applicable in areas such as quantitative finance, anomalous diffusion in physics, hydrology, and engineering systems with hereditary effects. Full article

This paper is concerned with a chemotaxis-competition system modeling the spatiotemporal evolution of two species that proliferate and compete according to Lotka–Volterra-type kinetics. We study the asymptotic behavior of solutions in the case of strong competition and show that they spatially segregate as the competition rate tends to infinity. Moreover, using a blow-up method, we obtain the uniform Hölder continuity of the solutions. Full article

We investigate the asymptotic behavior of a proposed ordinary differential equation (ODE) model for Genetic Toggle switches from Gardner et. al. and I. Rajapakse and S. Smale: ${\displaystyle \frac{dx}{dt}}={\displaystyle \frac{a}{1+y^m}}-x$ and ${\displaystyle \frac{dy}{dt}}={\displaystyle \frac{b}{1+x^n}}-y$ where $a,b,m,n>0$ and $x\left(t\right),y\left(t\right)\ge 0$. We also investigate the asymptotic behavior of the Euler discretization of this system: $x_{n+1}=a_1{x}_n+{\displaystyle \frac{b_1}{1+y_n^m}}=f(x_n,y_n)$ and $y_{n+1}=a_2{y}_n+{\displaystyle \frac{b_2}{1+x_n^n}}=g(x_n,y_n),$ where $1-h=a_1$, $1-k=a_2$, $ah=b_1$ and $bk=b_2$, $a_1,a_2\in (0,1)$ and $h,k>0$ are steps of discretizations. Here, x and y represent protein concentrations at a particular time in both genes and $a,b,m,n>0$, respectively, above. We will apply the theory of competitive maps to find the basins of attractions of different equilibrium points and period-two solutions of systems of difference equations. Full article

This article explores the oscillation properties of solutions for higher-order nonlinear neutral functional differential equations. This study sought to establish applicable oscillation conditions for nonlinear NDDEs when $\alpha \ne \beta$. Our analysis focused on identifying the key asymptotic characteristics of positive solutions in the noncanonical case. Based on these properties, we developed a multi-conditional criterion that guarantees oscillation, thereby extending previously established results for second-order equations. Unlike previous results, which involved arbitrary constants and were thus difficult to apply, the criteria derived in this work do not contain such constants, making them directly applicable. Moreover, to overcome the complexity introduced by the multiple constraints of this criterion, we propose a new and previously unreported simplified oscillation condition. The effectiveness of this simplified criterion is demonstrated through an illustrative example. Full article

Asymptotic solutions that can describe the incidence and reflection of waves have been used in various situations. They can also be applied to inverse problems and provide useful information in situations where a precise evaluation is required. However, the construction of standard asymptotic solutions requires higher regularity with respect to the boundaries of the observation target. This article proposes a “modified asymptotic solution” to overcome this weakness. To demonstrate its usefulness, it is applied to the analysis of the indicator function in the enclosure method for the inverse problem of the wave equation in a mixed-type medium. Full article

In this study, we consider a potential expressed as a hyperbolic-sine function aiming to achieve the energy eigenvalues in a closed form, that is, as an analytical expression. Based on this, the Schrödinger equation is constructed within the framework of non-relativistic quantum mechanics and is tackled by using the Asymptotic Iteration Method. The potential in question was previously addressed in the literature. As an alternative, we obtain the complete energy spectrum in a closed form for the single-well regime of the potential function, by way of the quasi-exact solvability where the system has analytical energy eigenvalues once a certain condition is met, or a relation between the potential parameters is satisfied. This is provided by the applicability of the Asymptotic Iteration Method to both quasi-exact and numerical solutions. Thus, the effects of the potential parameters on the energy spectrum can be seen separately. We conclude that the accuracy of the obtained closed-form energy spectrum is quite high as evidenced by the strong agreement with the numerically obtained ones. Furthermore, it is seen that this consistency improves as the energy level increases. The obtained analytical expression can also be used as a simple analytical model for vibrational spectrum of molecular systems described by anharmonic single-well potentials. Full article