Search Results (575)

This work investigates a sophisticated class of neutral fuzzy fractional functional differential equations (N3FDEs), where the fractional order $\alpha$ satisfies $0<\alpha \le 1$. We present a comprehensive analysis of the existence, uniqueness, and well-posedness of solutions under the generalized Hukuhara framework. First, we examine the existence and uniqueness of solutions under the generalized Hukuhara framework, providing an refined iterative formula for linear systems. We further verify the system’s well-posedness, proving that solutions remain stable and respond continuously to changes in initial data and parameters. Second, we introduce a novel spectral Vieta–Lucas projection method to approximate the solution. By leveraging the unique properties of Vieta–Lucas polynomials, we transform complex memory-dependent fuzzy equations into a streamlined algebraic system. Finally, numerical examples and error analysis show the method is accurate and efficient. Full article

The proper-time method for constructing perturbative dynamical gravitational fields is presented. Using the proper-time method, a perturbative analytical model of gravitational waves against the backdrop of an exact wave solution of Einstein’s equations in a Bianchi IV universe is constructed. To construct the perturbative analytical wave model a privileged wave coordinate system and a synchronous time function associated with the proper time of an observer freely moving in a gravitational wave are used. Reduction of the field equations, taking into account compatibility conditions, reduces the mathematical model of gravitational waves to a system of coupled ordinary differential equations for functions of the wave variable. Analytical solutions for the components of the gravitational wave metric have been found. The stability of the resulting perturbative solutions for the continuum domain of parameters is proven. The linear stability of the exact solution for a gravitational wave in the anisotropic Bianchi IV universe for the continuum domain of parameters is demonstrated. Full article

A novel two-stage procedure for approximating solutions of nonlinear systems is introduced. The scheme employs two evaluations of the vector function F together with a single Jacobian computation, followed by the resolution of two linear subproblems that share an identical coefficient matrix. This structure reduces the computational cost and enhances the adaptability of the method with respect to existing alternatives. The design of the algorithm is motivated by criteria relating efficiency to the total number of functional evaluations, ensuring that the resulting strategy achieves the optimal convergence order permitted within this framework. A proof of the local convergence order is provided, and its accuracy is supported by a series of experiments on distinct nonlinear models, including problems arising from differential equations. The numerical evidence confirms that the developed technique reaches the theoretical convergence rate and performs favorably when compared with other methods of equal order. Moreover, we examine the dynamical features of the related parametric variant, offering additional understanding of its stability properties and iterative behavior. Full article

Functionally graded porous beams are increasingly used in lightweight engineering structures, where thermal effects and material inhomogeneity significantly influence vibration behavior. In this study, the thermoelastic free vibration of functionally graded porous Euler–Bernoulli beams with temperature-dependent material properties is investigated by considering uniform and symmetric porosity distributions, together with uniform, linear, and nonlinear temperature fields. The governing equations are derived based on classical Euler–Bernoulli beam theory and solved using the Differential Transformation Method, while the accuracy of the semi-analytical formulation is verified through a Hermite-based finite element model. The results show that increasing temperature reduces the bending stiffness due to thermal axial forces and leads to a rapid decrease in natural frequency as the critical buckling temperature is approached. Increasing porosity generally decreases the natural frequency, although a slight increase may occur in symmetric distributions because of the accompanying reduction in mass density. The present study provides a computational framework for the thermo-vibration analysis of functionally graded porous beams in lightweight structural applications. Full article

This paper investigates the spatial asymptotic behavior of solutions to a class of nonlinear parabolic equations defined on an exterior region in ${\mathbb{R}}^3$. By constructing a suitable weighted energy functional and employing a fractional-order differential inequality technique, we establish a sharp Phragmén–Lindelöf type alternative: the solution either ceases to exist at a finite radial distance or decays to zero as the radial variable $r\to \infty$ when the power $p>2$. In the decay case, we derive explicit polynomial type decay estimates. The analysis is conducted in unbounded exterior domains where traditional compactness arguments are not applicable, extending previous studies on semi-infinite cylinders to more complex geometric settings. Our results reveal distinct spatial behaviors compared to those observed in linear or differently nonlinear parabolic problems and can be seen as a version of Saint-Venant principle in exterior regions. Full article

This study presents a systematic comparative benchmark between two distinct paradigms for solving nonlinear partial differential equations (PDEs): the data-driven Physics-Informed Neural Networks (PINNs) and the analytical Homotopy Analysis Method (HAM). We apply both methods to a unified family of canonical PDEs, the Burgers, Huxley, Fisher, Burgers–Huxley, and Burgers–Fisher equations, under identical problem setups, domain discretization, and validation metrics. PINNs incorporate physical laws directly into neural network training by minimizing a loss function that enforces PDE residuals, yielding physically consistent solutions even for strongly nonlinear problems. HAM provides approximate analytical solutions using a unified framework, and the same initial guess, auxiliary linear operator, and auxiliary function across all equations despite their distinct nonlinearities. The controlled, consistent application of both methods enables a fair, reproducible comparison across this equation family. The results provide a quantitative performance map under identical conditions, delineating when PINNs (high accuracy, long-term stability, and generalization capability) are preferable, versus when HAM (computational speed, short-term analytic approximation, and lower memory footprint) offers advantages. While the finite radius of convergence of the truncated HAM series is theoretically expected, our controlled comparison quantifies for the first time how this degradation varies across equation types, revealing that the choice between methods depends on specific problem requirements including error tolerance, available computational resources, and temporal horizon. The novelty lies not in solving each equation individually, but in deriving a performance taxonomy that systematically connects equation features (shocks, stiffness, and reaction–diffusion coupling) to optimal solver choice—providing previously unavailable, evidence-based guidance for the scientific computing community. This study establishes the first rigorous, controlled comparative benchmark between analytic and data-driven PDE solvers across a spectrum of nonlinearities, providing a reproducible baseline for future hybrid scientific machine learning solvers. Full article

This paper introduces a new explicit integration method for second-order ordinary differential equations (ODEs) commonly encountered in engineering applications. Traditionally, these problems are solved either by reformulating them as first-order systems to apply one-step methods such as Runge–Kutta schemes, or by using direct second-order approaches widely adopted in linear dynamics, including the generalized-α, central difference, and Newmark methods. The proposed method is derived from a Taylor series expansion truncated at the third derivative, resulting in a fully explicit algorithm that requires only one function evaluation per time step. Similar to Newmark’s formulation, it includes adjustable parameters that allow the user to balance accuracy and stability. For a specific parameter choice, the method exhibits convergence and stability properties comparable to those of the central difference scheme. An important advantage is that it remains explicit even when nonlinearities depend on first-derivative terms. The paper presents a theoretical analysis covering stability, local truncation error, spectral properties, numerical damping, and period elongation. The method is validated through four test cases from multibody dynamics, including linear and nonlinear problems. Results demonstrate that the Explicit Integration Grade 3 (EIG-3) method achieves accuracy comparable to existing explicit second-order integrators while significantly reducing computational cost, particularly in nonlinear applications. 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

The complete analytic solution of the time-dependent Vlasov–Boltzmann kinetic equation is used to describe selected problems in plasma physics within the framework of the quasi-linear approximation. These problems usually include the relaxation of plasma oscillations and the relaxation of beam instability. Our kinetic equation is a first-order partial differential equation. The method of characteristics allows us to solve it analytically, while fully preserving the entire time dependence. Using the obtained analytic expression for the distribution function, the paper shows that the indicated relaxation processes do not occur in the approximation considered. Full article

Time-fractional interface problems, found in heat transfer with discontinuous conductivities and fluid flows with surface tension forces, are challenging due to irregular interfaces and the history-dependent nature of fractional derivatives. This paper presents two numerical methods for simulating time-fractional double interface problems. The first method uses the Haar wavelet collocation technique, while the second relies on a meshless approach with radial basis functions. The fractional derivatives are replaced with the Caputo sense, the resulting first-order time derivatives are handled using the finite difference method, and the spatial operator is approximated using the two proposed methods. Gauss elimination is used to solve linear problems. Quasi-Newton linearization method is used for nonlinear problems. Both methods accommodate constant and variable coefficients, handling discontinuities and singularities in both solutions and coefficients. To evaluate the effectiveness of the proposed methods, numerical experiments are carried out. The accuracy of each method is quantified using the $L_{\infty }$ error norm, and a comparative analysis highlights the validity and advantages of the approaches. Moreover, the proposed schemes are rigorously analyzed to establish their stability, and the existence and uniqueness of the solutions. Full article

Score-based generative models, grounded in stochastic differential equations (SDEs), excel in producing high-quality data but suffer from slow sampling due to the extensive nonlinear computations required for iterative score function evaluations. We propose an innovative approach that integrates score-based reverse SDEs with kernel methods, leveraging the derivative reproducing property of reproducing kernel Hilbert spaces (RKHSs) to efficiently approximate the eigenfunctions and eigenvalues of the Fokker–Planck operator. This enables data generation through linear combinations of eigenfunctions, transforming computationally intensive nonlinear operations into efficient linear ones, thereby significantly reducing computational overhead. Notably, our experimental results demonstrate remarkable progress: despite a slight reduction in sample diversity, the sampling time for a single image on the CIFAR-10 dataset is reduced to an impressive 0.29 s, marking a substantial advancement in efficiency. This work introduces novel theoretical and practical tools for generative modeling, establishing a robust foundation for real-time applications. Full article

In this paper, we study a nonlinear system of sequential Caputo fractional differential equations equipped with coupled mixed multi-point boundary conditions. In particular, the boundary conditions involve the values of the unknown functions at the endpoints expressed as linear combinations of their values at several interior points, forming a closed system of relations. The existence of solutions is established by applying the Leray–Schauder alternative, while uniqueness is proved using Banach’s contraction principle. In addition, we investigate the Hyers–Ulam stability of the proposed system. Several examples are included to demonstrate the applicability of the theoretical results. Some special cases of the general problem are also discussed. Full article

This paper presents a real-time adaptive Linear Quadratic Regulator (LQR) control strategy for the rotary inverted pendulum. The state weighting matrix of the LQR cost function is continuously adapted online based on real-time tracking error, state dynamics, and sliding-mode-inspired robustness measures. Unlike conventional LQR controllers with fixed weighting matrices or hybrid schemes that apply sliding mode control directly to the control input, the proposed approach modulates the LQR cost function itself, enabling dynamic reshaping of controller behavior while preserving smooth control action. The real-time adaptive controller is implemented using a continuous-time Riccati differential equation solved online, making the method suitable for real-time deployment. Experimental validation is conducted on two Quanser QUBE-Servo 2 rotary inverted pendulum platforms under square, sinusoidal, and sawtooth reference trajectories. Performance is compared against a fixed-gain LQR controller using multiple quantitative metrics, including tracking error and control effort. Experimental results demonstrate substantial improvements in tracking accuracy, with reductions exceeding 70–90% in error metrics, while simultaneously achieving over 94% reduction in control effort. These findings verify that adaptive cost shaping provides an effective and practical mechanism for enhancing LQR performance in underactuated experimental systems. Full article

Postprandial plasma amino acid (AA) kinetics serve as essential indicators of digestive efficiency and systemic metabolic status in pigs. Traditional kinetic analysis relies on Non-Linear Least Squares (NLS) regression using compartmental models, yet these methods typically demand repeated blood sampling and precise initialization to ensure convergence. In this study, we developed a Physics-Informed Neural Network (PINN) framework by integrating mechanistic Ordinary Differential Equations (ODEs) directly into the deep learning loss function. The framework was evaluated using a benchmark dataset. Specifically, we performed a retrospective analysis by downsampling the original high-frequency data to simulate dense and sparse sampling strategies. The results demonstrate that while both models exhibit high fidelity under dense sampling, PINN maintains superior robustness and predictive accuracy under data-constrained conditions. Under the sparse sampling scenario, PINN reduced the Root Mean Square Error (RMSE) compared to NLS in key metabolic profiles, such as Methionine in the FAA group (p < 0.01) and Lysine in the HYD group (p < 0.05). Unlike NLS, which is sensitive to initial guesses, PINN successfully utilized physical laws as a regularization term to robustly solve the inverse problem, demonstrating superior parameter identification stability and predictive consistency under data-constrained conditions compared to NLS. We concluded that the PINN framework provides a reliable and consistent alternative for modeling the AA dynamics. In the future, it may be possible to reconstruct highly accurate physiological trajectories under optimized sparse sampling conditions. Full article

Fractional calculus approach is used to analyze the model of surfactant transport by anomalous diffusion and its adsorption on an interface in a liquid-liquid system. The anomalous diffusion is modeled by time-fractional partial differential equations in the bulk phases. The adsorption of surfactant is described by the corresponding time-fractional Neumann boundary conditions at the interface. The adsorption process is considered under mixed barrier-diffusion control, described by first-order ordinary differential equation, which relates the subsurface concentration with that on the interface. A second relation between these concentrations is derived in terms of a fractional equation by application of Laplace transform technique. By combining both relations the subsurface concentration is eliminated and a single multi-term fractional ordinary differential equation for the surfactant concentration on the interface is derived. Different adsorption kinetic models are considered. In the case of Henry adsorption isotherm the model is linear and possesses analytical solution in terms of multinomial Mittag-Leffler functions. In the cases of Volmer and van der Waals adsorption isotherms nonlinear differential equations of fractional order are obtained. They are reformulated in equivalent integral form, which is used for computer simulation of the process of adsorption. Numerical results are presented and compared with analytical asymptotic predictions. Full article