Search Results (790)

The sign of the maximal almost sure Lyapunov exponent determines the stability of stochastic systems, while its numerical computation for three-dimensional linear Stratonovich stochastic differential equations remains challenging due to the failure of classical two-dimensional strategies. The spherical angular motion of 3D systems produces a Fokker–Planck equation with intractable mixed partial derivatives, preventing conventional analytical solutions. This paper develops a unified computational framework for three-dimensional linear Stratonovich stochastic systems using analytical derivation for degenerate cases and physics-informed neural network (PINN) approximation for general non-degenerate scenarios. For degenerate systems, we reduce the coefficient matrix to a lower triangular form via orthogonal transformation and establish tight upper bounds based on the logarithmic growth property of the Wiener process, yielding closed-form expressions for the maximal almost sure Lyapunov exponent under all parameter sign configurations. For non-degenerate systems, we reformulate the Fokker–Planck equation in spherical coordinates and construct a customized PINN with trigonometric encoding to enforce periodic boundary conditions. The network is trained by joint loss functions of equation residuals, boundary constraints and normalization consistency, and the converged stationary density is substituted into the Furstenberg–Khasminskii formula to calculate the exponent via Gauss–Legendre quadrature. Monte Carlo simulations confirm the accuracy and robustness of the proposed method, which reliably identifies the sign of the maximal almost sure Lyapunov exponent even in near-critical regimes. Numerical experiments on a 3D stochastic Hopf bifurcation model show that noise negatively shifts the bifurcation point, with the offset linearly proportional to the squared noise intensity. This work extends Lyapunov stability analysis from two-dimensional to three-dimensional linear Stratonovich stochastic systems, offering an effective tool for stability evaluation of general three-dimensional stochastic dynamical models. 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

We develop a systematic framework for incorporating perturbative correction terms into classical finite difference schemes for Allen–Cahn type stochastic partial differential equations. Three distinct correction approaches are introduced, conceptually motivated by perturbative quantum field theory, quantum coherent state evolution, and WKB (Wentzel–Kramers–Brillouin) barrier penetration theory. These quantum-inspired perturbative method (QIPM) corrections function as classical perturbations executing entirely on conventional hardware; quantum-mechanical formalism serves only as a design principle for constructing specific functional forms of correction terms. The primary novelty of this work lies in (i) a generic convergence-preservation theorem establishing sufficient conditions on correction magnitude for any perturbative correction to maintain the base scheme’s accuracy order, and (ii) a systematic translation methodology from quantum-mechanical analogies to explicit, implementable finite difference corrections with rigorous parameter bounds. Through convergence analysis, we demonstrate that appropriately parametrized corrections preserve the accuracy of the underlying numerical scheme, provided the solution possesses sufficient regularity and the parabolic scaling constraint $\Delta t=\mathcal{O}\left(h^2\right)$ holds. Numerical experiments on a spatially discretized domain over a finite time horizon using spatially correlated noise reveal that the anharmonic oscillator correction achieves exceptional accuracy with modest computational overhead, while the amplitude encoding correction provides intermediate accuracy with negligible timing cost. The tunneling-inspired correction exhibits higher error for smooth initial conditions, indicating strong problem-dependence. While these methods enhance accuracy in specific scenarios, genuine speedups on classical hardware are not achieved. The primary value lies in establishing systematic methodologies for perturbative correction design and developing theoretical foundations for future hybrid computational approaches. Full article

In this study, we utilize the ${\varphi }^6$-model expansion method to derive a diverse set of Jacobi elliptic function solutions for the conformable resonant Nonlinear Schrödinger Equation (NLSE) with parabolic law nonlinearity. As the modulus of the Jacobi elliptic functions approaches 1 and 0, the solutions transform into hyperbolic and trigonometric functions, respectively. This methodology yields various exact traveling wave solutions, including kink solitons, singular solitons, periodic solutions, and singular periodic solutions. Notably, this work represents the first investigation into identifying Jacobi elliptic function solutions for the conformable resonant NLSE. These results enhance the understanding of the nonlinear dynamical properties intrinsic to the NLSE. We use graphical illustrations to highlight the dynamical features of the solutions. Moreover, our approach showcases versatility in addressing other nonlinear partial differential equations, offering insights applicable to nonlinear optics, fluid dynamics, and quantum physics. Full article

This paper investigates the optimal control problem of time-delay backward doubly stochastic systems under partial information. Partial information widely exists in practical control systems due to monitoring constraints, communication delays, and data acquisition costs. Combined with inherent system time delays, it greatly complicates state estimation and decision-making, which requires research. A new type of anticipated backward doubly stochastic differential equations is introduced to describe the system dynamics. Using stochastic analysis and the variational methods, the corresponding maximum principle for optimal control is derived. Furthermore, a verification theorem is established that provides rigorous sufficient optimality conditions: any admissible control satisfying the necessary conditions, along with reasonable convexity assumptions, indeed optimizes the cost functional, thereby bridging the gap between necessary and sufficient optimality criteria. As an application, we solve a time-delay linear-quadratic optimal control problem and obtain explicit analytical expressions; the results demonstrate the validity of the established theoretical framework. Full article

The electro-oxidation of persistent organic pollutants such as 2-chlorophenol (2-CPh) using boron-doped diamond (BDD) electrodes offers a promising wastewater treatment route, yet conventional mechanistic models (e.g., CFD) suffer from prohibitive computational costs. This study develops a hybrid physics-informed neural network (PINN) to model the electro-oxidation of 2-CPh in a flow-by reactor coupled with a continuous stirred tank under batch recirculation mode. The PINN integrates a diffusion–convection partial differential equation with a lumped-parameter ordinary differential equation for the tank, embedding physical constraints directly into the loss function. The model was trained on simulated data generated from a previously validated parametric model and optimized using a systematic hyperparameter grid search. The PINN achieved excellent agreement with experimental data, yielding a coefficient of determination (R²) of 0.9927, a mean square error of 0.0009, and a root mean square error of 0.0294—outperforming both the CFD and parametric models in accuracy. Sensitivity analysis revealed that the apparent kinetic constant is the most influential parameter (normalized sensitivity of 14.20). While the CFD model required 42 days and the parametric model 8 s, the PINN achieved a balanced trade-off with a runtime of 7.36 h. We conclude that the PINN provides a highly accurate, computationally feasible surrogate model suitable for integration into digital twins and real-time control frameworks for electrochemical wastewater treatment. Full article

This paper investigates the optimal harvesting of a nonlinear, size-structured population governed by a first-order transport equation with nonlocal environmental crowding feedback and exogenous inflow. First, we establish finite-horizon well-posedness for the controlled state system in an $L^1$ framework, proving the existence, uniqueness, positivity, and continuous dependence of weak solutions. Second, we show that the infinite-dimensional stationary problem reduces exactly to a scalar nonlinear closure equation, yielding existence and conditional uniqueness results for stationary states. Within this equilibrium framework, we distinguish the persistence of the forced system from intrinsic demographic self-replacement and introduce size-continuous per-recruit and spawning-potential diagnostics. Finally, we formulate a partial differential equation (PDE)-constrained optimal harvesting problem. Under a compactness assumption on the control-to-state map, we establish the existence of optimal controls. We then formally derive a Pontryagin-type first-order optimality system for the harvesting problem. The variation of the nonlocal environmental feedback produces a coupled integral source term in the adjoint equation. The associated pointwise maximization condition yields a bang–bang harvesting structure, while a monotone size-threshold policy is shown to require an additional single-crossing assumption on the switching function. These hypotheses are illustrated using a fisheries model with density-dependent von Bertalanffy growth. Full article

In the present work, we introduce and study a new two-parameter generalization of Legendre-based Appell polynomials, defined through an explicit representation that unifies classical Legendre structures with the Appell polynomial framework. Starting from a generating function, we derive a three-term recurrence relation, a degree-lowering operator, an integro-partial degree-raising operator, and a corresponding integro-partial differential equation satisfied by the new family. A determinant representation is established via Cramer’s rule applied to the Cauchy-product expansion of the generating function. Several subfamilies of independent interest arise naturally as special cases, namely, Legendre-based Hermite–Frobenius–Euler polynomials, Legendre-based Miller–Lee polynomials, and both the probabilist’s and physicist’s variants of Legendre-based bi-variate Hermite polynomials. For each subfamily we record the corresponding recurrence relations, shift operators, differential equations, and determinant forms, and we illustrate the behavior of selected members through three-dimensional surface plots and real-root distribution diagrams. The framework presented here extends several constructions available in the recent literature and points to natural directions for future work, including connections with q-series, combinatorial identities, and symbolic-computation methods, which are outlined in the concluding section. Full article

Fractional derivatives introduce an effective mathematical structure to describe memory effects, long-range interactions, and anomalous transport processes that are not well represented by the traditional integer-order models. This paper presents the unidirectional fractional longitudinal wave equation as a governing equation where a model is proposed to explain the steady wave propagation of solitary waves in a magneto-electro-elastic circular rod. Magneto-electro-elastic substances are a groundbreaking category of advanced functional materials with tremendous nanotechnology and biomedical engineering prospects because of their effective multi-field energy conversion and temperature responsiveness. In order to solve this complicated fractional nonlinear equation, we introduce a new computation-analysis approach: the Riccati subequation neural network method. This hybrid solution is a synergistic combination of an analytical solution structure and a neural network structure consisting of input, hidden, and output layers, with interconnection between neurons through weighted connections and activation functions. It is important to note that every neuron in the first hidden layer is coupled to the solutions of the Riccati equation, and this allows the systematic use of the new trial functions. With the suggested method, analytical solutions are obtained for the spacetime fractional partial differential equations of the unidirectional fractional longitudinal wave equation in the exact form of trigonometric, hyperbolic, and rational functions. This paper is the first attempt to combine the Riccati subequation method with a neural network model, which has given rise to new types of solitary wave solutions. The three-dimensional, two-dimensional, and contour plots are used to visualize the dynamic nature of these solutions and to display the rich nonlinear wave behavior. The effectiveness and the robustness of the implemented technique is not only proven through our findings but also provides more profound information about the nonlinear wave phenomena in the advanced multifunctional materials, which can inform future developments in energy harvesting and the design of biomedical devices. Full article

This study examines how institutional pressures influence the adoption of social sustainability practices in habitat systems within the construction sector. Drawing on Institutional Theory, the research analyzes the differentiated effects of coercive, mimetic, and normative pressures, as well as the mediating role of organizational awareness. Data were collected through a digital survey administered between February and March 2026 to 102 professionals linked to construction and habitat development projects in Mexico, including architects, civil engineers, valuators, and related specialists. The proposed model was evaluated using Partial Least Squares Structural Equation Modeling (PLS-SEM). The results show that coercive pressures constitute the only statistically significant institutional mechanism affecting organizational awareness ($\beta$ = 0.310; p = 0.043), while mimetic and normative pressures do not exhibit significant effects. Furthermore, organizational awareness strongly explains the adoption of social sustainability practices ($\beta$ = 0.739; p < 0.001), which, in turn, is strongly associated with sustainable habitat outcomes ($\beta$ = 0.711; p < 0.001). The model achieved moderate predictive power, with $R^2$ values of 0.449 for awareness, 0.546 for adoption, and 0.505 for sustainable habitat systems. The findings contribute to institutional theory by demonstrating that institutional mechanisms operate asymmetrically in emerging contexts and that organizational awareness functions as a key explanatory mechanism linking external pressures with sustainability outcomes. The study also provides practical implications for urban governance, regulatory design, and socially sustainable habitat planning. Full article

Electron emission from hydrogen atoms induced by antiproton impact at intermediate energies is investigated using the one-centre Basis Generator Method within a semi-classical impact-parameter framework. The formulation employs a single-centre expansion of the time-dependent Schrödinger equation with a pseudostate basis consisting of hydrogenic orbitals acted upon by powers of a Yukawa-regularized potential, providing a compact and effective representation of the electronic continuum. Ionization probabilities are obtained by projecting the time-evolved wavefunction onto Coulomb continuum states, from which energy-differential cross sections (EDCS) are extracted. Exponential piecewise functions are constructed to interpolate between the pseudostate eigenenergies, yielding smooth EDCS profiles for each partial wave. The total EDCS, reconstructed by summing over all partial-wave contributions, exhibits good agreement with results from other pseudostate-based approaches. Full article

This work addresses the boundary disturbance attenuation problem for a flexible beam governed by a fourth-order partial differential equation. A boundary disturbance observer based on a radial basis function neural network is proposed to achieve high-accuracy online estimation of disturbances without prior knowledge of the disturbance dynamics. In addition, a boundary feedback controller acting only at the fixed end is designed. The control objectives are to ensure accurate tracking of the desired angular position, suppress elastic vibrations, and attenuate the influence of unknown time-varying boundary disturbances. By constructing a Lyapunov functional, the stability of the closed-loop system is established. Numerical simulations demonstrate the effectiveness of the proposed observer and control law. Full article

The new projective solutions methods (PSMs) for solving the Stokes, Oseen, and Brinkman flow problems are presented in this paper. They automatically satisfy the governing equations and are therefore Trefftz-type methods. Utilizing the third-order formulation and three-dimensional analytic functions, we derive a meshless Trefftz-type method to solve three-dimensional Stokes flow problems. The Oseen and Brinkman equations are transformed into four coupled third-order/first-order partial differential equations. The projective-type particular solution (PTPS) is obtained via a projective function in terms of the projective variable; the third-order ordinary differential equations (ODEs) with constant coefficients are derived to determine the projective functions. The Trefftz-type PSM is extremely accurate, because the governing equations (including the incompressibility condition) are implemented automatically. For the Brinkman equations, the general solutions of velocity and pressure are presented by using the Helmholtz function and a harmonic function, whose corresponding Trefftz-type numerical method is developed. Upon comparison with the method of fundamental solutions (MFS), the new methods exhibit some advantages, including lower condition numbers, faster convergence, and better accuracy. We also apply the Trefftz-type PSM to solve the exterior problem of the Stokes equations, where the velocity tends to zero at infinity. Full article

After more than a century of unremitting efforts by scholars, nonlinear analysis has found widespread and important applications in many fields that are at the core of many branches of pure and applied mathematics, including functional analysis, fixed point theory, nonlinear ordinary and partial differential equations, variational analysis, dynamical system theory, control theory, convex analysis, nonsmooth analysis, critical point theory, nonlinear optimization, fractional calculus and its applications, probability and statistics, mathematical economics, data mining, signal processing, biological engineering, electronic networks, electromagnetic theory, and so forth [...] Full article

Accurate estimation of the internal spatial-temporal temperature distribution is crucial for the safety and performance management of lithium-ion batteries. However, traditional lumped parameter models overlook spatial gradients, while numerical methods for partial differential equations (PDEs) incur high computational costs. This paper proposes a hard constraint physics-informed neural network (HCPINN) framework for the real-time reconstruction of the axial temperature field in 18,650 cylindrical batteries. By restructuring the neural network’s solution space through distance functions, the Robin boundary conditions are strictly embedded as hard constraints, ensuring exact satisfaction of the prescribed Robin boundary conditions within the mathematical model and eliminating boundary loss terms. An electro-thermal coupled model considering the Arrhenius effect and state-of-charge (SOC) dependent internal resistance is integrated into the loss function to capture the nonlinear heat generation dynamics. Experimental validation across discharge rates from 1C to 4C demonstrates that the HCPINN achieves high estimation accuracy with a mean absolute error (MAE) below 0.34 °C. Furthermore, by leveraging the continuous differentiability of the model, this study quantifies the evolution of spatial temperature gradients and reveals the ideal heat transfer coefficients required for thermal equilibrium are inverted, providing a quantitative basis for the design of advanced battery thermal management systems (BTMS). Full article