Search Results (534)

In this paper, we investigate the (1 + 1)-dimensional nonlinear truncated M-fractional FitzHugh–Nagumo model. The main objective is to analyze the dynamical behavior and obtain exact solutions for the model. First, a fractional transformation is applied to convert the governing partial differential equation into an ordinary differential equation. Subsequently, a Galilean transformation is employed to reduce the resulting equation to a dynamical system. The bifurcation structure and chaotic dynamics of the model are then examined. The presence of chaos is further confirmed through the phase portrait, basin of attraction, return map, Lyapunov exponent, permutation entropy, Poincaré map, power spectrum, attractor, fractal dimension, multistability, time analysis, and recurrence plot. In addition, the sensitivity of the system to the initial conditions is analyzed. Finally, exact solutions for the model are constructed using the unified Riccati equation expansion method. The obtained results are illustrated using two-dimensional, three-dimensional, and contour plots. Full article

In this paper, we investigate the spectrum distribution and asymptotic behavior of an M/$G^B$/1 bulk service queueing system. In this system, the server processes customers in batches of a fixed maximum capacity B, and the time required to serve a batch is governed by a general distribution with a service rate function $\eta (·)$, which determines the instantaneous probability of service completion. The system dynamics are described by an infinite set of partial integro-differential equations. First, by introducing the probability generating function and employing Greiner’s boundary perturbation method, we establish that the time-dependent solution (TDS) of the system converges strongly to its steady-state solution (SSS) in the natural Banach state space. To this end, when the service rate $\eta (·)$ is a bounded function, we prove that zero is an eigenvalue of both the system operator and its adjoint operator, with geometric multiplicity one. Moreover, we show that every point on the imaginary axis except zero belongs to the resolvent set of the system operator. Second, we analyze the spectrum of the system operator on the left real axis. When the service rate $\eta (·)$ is constant and the fixed maximum capacity B equals 2, we apply Jury’s stability criterion for cubic equations to demonstrate that the system operator possesses an uncountably infinite number of eigenvalues located on the negative real axis. Additionally, we prove that an open interval near zero on the left real axis is not part of the point spectrum of the system operator. Consequently, these results imply that the semigroup generated by the system operator is not compact, eventually compact, quasi-compact, or essentially compact. Full article

We investigate a stochastic coupled nonlinear Schrödinger (Manakov-type) system for option price and volatility wave fields within the Ivancevic adaptive-wave option-pricing paradigm, and derive exact wave families together with statistical diagnostics of the resulting dynamics. This system combines behavioral market effects with classical efficient-market dynamics and incorporates a controlled stochastic volatility component. Randomness in both the option price and volatility is incorporated via white noise, and a system of stochastic partial differential equations (PDEs) is developed that governs the joint evolution of option prices and stock price volatility. We derive advanced solutions of the proposed system using a newly created methodology. The obtained solutions are expressions of cnoidal, snoidal, dnoidal, hyperbolic, trigonometric, and exponential functions. The stochastic dynamical investigation, together with the statistical measures are presented. The autocorrelation function (ACF) of squared returns for the obtained analytical solutions is demonstrated to show distinct differences in second-order temporal dependence, while asymmetries in the temporal evolution of the fluctuations are depicted via leverage correlation (LC). The probability distribution function (PDF) dynamics of the soliton solutions illustrate prominent temporal variability and non-stationary statistical dynamics. Differences in dynamical coupling between the two components of the considered system are presented via phase velocity cross-correlation analysis and are supported by phase difference dynamics visualizations. The strength and structure of coupling between components are displayed via the amplitude cross-correlation function. Mean amplitude dynamics and variance as a function of noise intensity $\sigma$, provide a systematic influence of stochastic forcing on their energy and a quantitative measure of stochastic dispersion of soliton solutions. All the results are displayed in 3D and 2D graphs of the stochastics and statistical dynamics of the obtained solutions. Full article

An inhomogeneous thin internal stratum sometimes exists between two dissimilar materials, which is usually caused by non-uniform thermal distribution, interaction of different media, diffusion impurity or material degeneration and damage. In this paper, it is considered as a functional graded (FG) piezoelectric material in surface acoustic wave devices, and we investigate its effect on Love wave propagation within the framework of the linear piezoelectric theory. Correspondingly, the power series technique is presented and applied to solve the dynamic governing equations, i.e., two-dimensional partial differential equations with variable coefficients, with the convergence and correctness being proved. In this method, the material coefficients can change in random functions along the thickness direction, which reveals the generality of this method to some extent. As the numerical case, the elastic coefficient, piezoelectric coefficient, dielectric permittivity, and mass density change in the linear form but with different graded parameters, and the influence of material inhomogeneity on the Love wave propagation is systematically investigated, including the phase velocity, electromechanical coupling factor, and displacement distribution. In addition, the FG piezoelectric material caused by piezoelectric damage and material bonding is discussed. Numerical results demonstrated that both piezoelectric damaged and material bonding can make the higher modes appear earlier for the electrically open case, decrease the initial phase velocity, and limit the existing region of the fundamental Love mode for the electrically shorted case. The qualitative conclusions and quantitative results can provide a theoretical guide for the structural design of surface wave devices and sensors. Full article

This research addresses a structural cybernetic anomaly within strategic management precipitated by the integration of artificial intelligence into the organizational core. Traditional paradigms, specifically the resource-based view and the dynamic capabilities framework, operate under closed-system, first-order cybernetic assumptions that fail to capture the dissipative nature of algorithmic agents. By conceptualizing the enterprise as a complex adaptive system operating far from thermodynamic equilibrium, this study introduces the theory of dynamic cognitive advantage. Grounded in second-order cybernetics, the framework posits that competitive differentiation emerges from the historical, recursive, structural coupling of human semantic intent and machine syntactic processing. This research formalizes this co-evolutionary dynamic utilizing coupled non-linear differential equations and time decay integrals. Furthermore, it operationalizes the central mechanism of this capability—the cognitive flywheel—and proposes a fractal governance architecture to mitigate systemic vulnerabilities such as automation bias. To transition these propositions into management science, a proposed mixed-methods empirical research agenda is presented. It outlines a future partial least squares–structural equation modeling (PLS-SEM) approach to test the mediating role of the cognitive flywheel and the moderating effect of fractal governance on organizational resilience. This research provides a mathematically formalized, empirically testable architecture for navigating the artificial intelligence economy. Full article

In this study, we propose a compartmental mathematical model that considers two interacting populations (citrus plants and insect vectors) and investigate the transmission dynamics of Huanglongbing in citrus crops. This disease is caused by the bacterium Candidatus Liberibacter asiaticus and is vectored by the psyllid Diaphorina citri. The disease is modeled under the following three main assumptions: there is vital dynamics with constant recruitment rates of citrus plants, the force of infection in both populations is a spatially dependent function varying with geographic location, and there is a spatial displacement of the vectors. In the main results of the paper, we formulate a coupled ordinary and partial differential equation system with initial and zero flux boundary conditions, establish the existence and uniqueness of solutions to the proposed model by applying semigroup theory, and introduce a numerical approximation of the system. Moreover, we develop a stability and persistence analysis. From the analytical point of view, we calculate the basic reproduction number $R_0$ and prove three facts: the disease-free equilibrium is globally asymptotically stable when $R_0<1$; the disease-free equilibrium is globally asymptotically stable when $R_0>1$; and the hybrid system exhibits uniform persistence of infection when $R_0>1$. In addition, we present some numerical examples. Full article

This paper investigates the optimal investment and reinsurance problem for insurers in a financial market with contagion risk. The prices of risky assets are assumed to follow a jump–diffusion model, where the jump component is driven by a multidimensional dynamic contagion process with diffusion (DCPD). This process simultaneously captures jumps triggered by endogenous and exogenous excitations, effectively characterizing the dynamic contagion effects arising from the joint influence of multiple factors in financial markets. The insurer aims to maximize a mean-variance (MV) utility function by purchasing per-loss reinsurance and investing the surplus in the contagion financial market. By solving the extended Hamilton–Jacobi–Bellman (HJB) equations, we derive the time-consistent equilibrium investment and reinsurance strategies, as well as explicit expressions for the equilibrium value function. These results are characterized by two nonlocal partial differential equations (PDEs), whose probabilistic solutions are obtained through the Feynman–Kac formula. Finally, numerical experiments illustrate how equilibrium strategies respond to changes in contagion intensity and confirm the effectiveness of the proposed model. Full article

This study examines a nonlinear partial differential equation, namely the (3+1)-dimensional Boussinesq-type equation. To explore this model, three versatile analytical approaches are applied: the Exp-function method, the Kudryashov method, and the Riccati equation method. Using these techniques, a range of exact analytical solutions is derived, exhibiting diverse structural forms such as periodic, kink-type, rational, and trigonometric solutions. The analysis reveals the rich dynamical behavior of the equation and demonstrates its effectiveness in modeling a variety of nonlinear wave phenomena across different physical contexts. Several of the obtained solutions are illustrated through graphical representations for better interpretation. The results include hyperbolic, trigonometric, and rational function solutions, along with a sensitivity analysis. To highlight the physical relevance of the findings, suitable parameter values are selected, and the corresponding wave behaviors are visualized using three-dimensional and contour plots generated with Maple 2024. Overall, the study provides valuable insights into the mechanisms underlying the generation and propagation of complex nonlinear phenomena in fields such as fluid dynamics, optical fiber systems, plasma physics, and ocean wave transmission. Full article

We present Hyperbolic Symmetric Hypermodular Neural Operators (ONHSH), a novel operator learning framework for solving partial differential equations (PDEs) in curved, anisotropic, and modularly structured domains. The architecture integrates three components: hyperbolic-symmetric activation kernels that adapt to non-Euclidean geometries, modular spectral smoothing informed by arithmetic regularity, and curvature-sensitive kernels based on anisotropic Besov theory. In its theoretical foundation, the Ramanujan–Santos–Sales Hypermodular Operator Theorem establishes minimax-optimal approximation rates and provides a spectral-topological interpretation through noncommutative Chern characters. These contributions unify harmonic analysis, approximation theory, and arithmetic topology into a single operator learning paradigm. In addition to theoretical advances, ONHSH achieves robust empirical results. Numerical experiments on thermal diffusion problems demonstrate superior accuracy and stability compared to Fourier Neural Operators and Geo-FNO. The method consistently resolves high-frequency modes, preserves geometric fidelity in curved domains, and maintains robust convergence in anisotropic regimes. Error decay rates closely match theoretical minimax predictions, while Voronovskaya-type expansions capture the tradeoffs between bias and spectral variance observed in practice. Notably, ONHSH kernels preserve Lorentz invariance, enabling accurate modeling of relativistic PDE dynamics. Overall, ONHSH combines rigorous theoretical guarantees with practical performance improvements, making it a versatile and geometry-adaptable framework for operator learning. By connecting harmonic analysis, spectral geometry, and machine learning, this work advances both the mathematical foundations and the empirical scope of PDE-based modeling in structured, curved, and arithmetically. Full article

The present work is focused on a predator–prey model with the Holling type II interaction function, which is influenced by diffusion, advection and nonlinear reaction effects. Firstly, we show that the solutions of this dynamical model are bounded and unique. Secondly we use the Lyapunov function and then show that the equilibrium points are globally stable. Thirdly, we obtain the solution profile when the diffusion coefficient is small. For this purpose we introduce self-similar structures to convert the nonlinear partial differential equations into nonlinear ordinary differential equations and then use the singular perturbation technique to solve these equations. Fourthly, we use the Hamiltonian and Lighthill’s technique to obtain upper stationary solutions for a small coefficient of the advection term. Lastly, we consider a large diffusion coefficient and obtain the asymptotic profiles of nonstationary solutions with the help of nonlinear point scaling. Full article

Physics-informed neural networks (PINNs) solve partial differential equations (PDEs) by embedding physical conditions as soft penalties into the loss function. However, the coexistence of multiple loss components often leads to gradient conflicts, degrading convergence and solution accuracy. To address this issue, we propose a dynamic domain–gradient loss reweighting PINN (DDR-PINN). The proposed method introduces a dual-residual reweighting mechanism based on gradient variations, where adaptive weights are derived from the L2 norm of the dot product between loss gradients and residuals. These weights are further normalized through a nonlinear hyperbolic tangent transformation, enabling dynamic and balanced reweighting of interior, initial, and boundary domain losses throughout training. Extensive numerical experiments on PDEs with both Dirichlet and Neumann boundary conditions demonstrate that the DDR-PINN consistently outperforms the standard PINN, APINN, and VI-PINN with the fewest trainable parameters. Full article

As a mesh-free solving paradigm, Physics-Informed Neural Networks (PINNs) demonstrate potential in both forward and inverse problems by embedding physical equations into the loss function. However, they still face challenges in capturing the spatiotemporal evolution of complex physical processes. When applied to time-dependent complex flows, such as high-Reynolds-number cylinder flow, they often rely on supervised data, which is frequently difficult to obtain accurately in practice. To address these issues, this paper proposes a novel unsupervised solving framework—the Adaptive Hard-Constraint Physics-Informed Neural Network (AHC-PINN). This method integrates an adaptive sampling mechanism based on partial differential equation residuals with a hard-constraint strategy. By dynamically evaluating the contribution of collocation points to the loss and incorporating analytically embedded boundary constraints, it directs the network training entirely toward solving the governing equations. Using two-dimensional unsteady cylinder flow as a validation case, experimental results show that AHC-PINN significantly improves the prediction accuracy of wake evolution under unsupervised conditions. Its performance surpasses that of traditional soft-constraint PINNs by an order of magnitude and is even superior to methods using sparse supervised data. Furthermore, through analysis of the PDE loss and gradient distribution, the study explicitly identifies the impact of large-gradient regions on PINN training stability and prediction accuracy, providing a basis for subsequent optimization. Full article

When solving partial differential equations (PDEs), traditional Physics-Informed Neural Networks (PINNs) often encounter difficulties in capturing critical physical features and addressing information bias between subdomains. To overcome these limitations, this paper proposes a Dual Adaptive Domain Decomposition Physics-Informed Neural Network (DADD-PINN). The core of this method lies in the construction of a dual-driven architecture that facilitates intra-subdomain feature extraction and inter-subdomain feature coordination. Within each subdomain, the solver’s precision is significantly enhanced by integrating a multi-criterion adaptive sampling strategy with a dynamic weighting mechanism. Experimental results demonstrate that DADD-PINN reduces the optimal $L^2$ error by 1–2 orders of magnitude compared to existing baselines. The model exhibits superior generalization and robustness across various physical fields, offering a new route toward accurate and efficient solutions for complex PDEs. Full article

The (2+1)-dimensional Boussinesq equation is a fundamental model in nonlinear wave theory, governing shallow-water wave propagation, coastal dynamics in ocean engineering, and long waves in geophysical fluid systems such as atmospheric and oceanic currents. We present a novel neural network symbolic computation framework that seamlessly integrates neural architectures for powerful function approximation with symbolic manipulation for exact algebraic resolution, eliminating the need for bilinear transformations and thereby substantially reducing computational complexity. Applying this framework, we derive five previously unreported exact analytical solutions using carefully designed neural network configurations and probe functions. These solutions provide valuable tools for modeling ocean internal waves, coastal engineering simulations, and nonlinear optical pulse dynamics. In practice, the method delivers faster and more accurate simulations, improving engineering design and environmental prediction capabilities. By synergistically combining neural networks with symbolic computation, our approach surpasses traditional numerical methods and physics-informed neural networks in both accuracy and efficiency, opening new avenues for solving complex nonlinear partial differential equations. Full article

This study presents the mathematical formulation of a symmetry-compact three-step algorithm (TSA) for the numerical computation of the spatio-temporal generalized FitzHugh–Nagumo equation (FHNE), a class of one-dimensional time-dependent initial-boundary value partial differential equations. The proposed symmetry-compact TSA is constructed using the Lagrange polynomial as the basis function, yielding a structurally balanced and computationally compact formulation with an inherent symmetry that facilitates automatic step-size adaptation over the integration interval. The symmetry-compact nature of the formulation enhances numerical stability while maintaining a reduced computational footprint, thereby improving both accuracy and efficiency when compared with existing numerical schemes. Prior to the application of the TSA, the FHNE is discretized in space, resulting in a system of ordinary differential equations suitable for time integration. Rigorous analyses of the stability and convergence properties of the symmetry-compact TSA are carried out to establish the reliability and robustness of the method. The performance of the proposed algorithm is quantitatively assessed using absolute error, maximum error, root mean square error, and central processing unit time for selected spatio-temporal test cases of the FHNE. The numerical results and corresponding solution profiles clearly demonstrate that the symmetry-compact TSA delivers superior accuracy, enhanced computational efficiency, and improved stability characteristics relative to existing methods, particularly in the presence of stiffness and chaotic dynamics. Full article