Search Results (195)

Fractional differential equations have emerged as a prominent focus of modern scientific research due to their advantages in describing the complexity and nonlinear behavior of many physical phenomena. In particular, when considering problems with initial-boundary value conditions, the solution of nonlinear fractional differential equations becomes particularly important. This paper aims to explore the fractional soliton solutions for the three-component fractional nonlinear Schrödinger (TFNLS) equation under the zero background. According to the Lax pair and fractional recursion operator, we obtain fractional nonlinear equations with Riesz fractional derivatives, which ensure the integrability of these equations. In particular, by the completeness relation of squared eigenfunctions, we derive the explicit form of the TFNLS equation. Subsequently, in the reflectionless case, we construct the fractional N-soliton solutions via the Riemann–Hilbert (RH) method. The analysis results indicate that as the order of the Riesz fractional derivative increases, the widths of both one-soliton and two-soliton solutions gradually decrease. However, the absolute values of wave velocity, phase velocity, and group velocity of one component of the vector soliton exhibit an increasing trend, and show power-law relationships with the amplitude. Full article

With the widespread integration of Artificial Intelligence-Generated Content (AIGC) into e-commerce platforms, understanding how users perceive, evaluate, and respond to such content has become a critical issue for both academia and industry. This study examines the influence mechanism of AIGC Content Quality (AIGCQ) on users’ Purchase Intention (PI) by constructing a cognitive model centered on Trust (TR) and Perceived Risk (PR). Additionally, it introduces two moderating variables—Ethical Concern (EC) and Perceived Platform Responsibility (PLR)—to explore higher-order psychological influences. The research variables were identified through a systematic literature review and expert interviews, followed by structural equation modeling based on data collected from 507 e-commerce users. The results indicate that AIGCQ significantly reduces users’ PR and enhances TR, while PR negatively and TR positively influence PI, validating the fundamental dual-pathway structure. However, the moderating effects reveal unexpected complexities: PLR simultaneously amplifies both the negative effect of PR and the positive effect of TR on PI, presenting a “dual amplification” pattern; meanwhile, EC weakens the strength of both pathways, manifesting a “dual attenuation” effect. These findings highlight the nonlinear cognitive mechanisms underlying users’ acceptance of AIGC, suggesting that PLR and EC influence decision-making in more intricate ways than previously anticipated. By uncovering the unanticipated patterns in moderation, this study extends the boundary conditions of the trust–risk theoretical framework within AIGC contexts. In practical terms, it reveals that PLR acts as a “double-edged sword,” providing more nuanced guidance for platform governance of AI-generated content, including responsibility frameworks and ethical labeling strategies. Full article

For a non-conservative nonlinear oscillator (NCNO) having a periodic solution, the existence of the first integral is a certain symmetry of the nonlinear dynamical system, which signifies the balance of kinetic energy and potential energy. A first-order nonlinear ordinary differential equation (ODE) is used to derive the first integral, which, equipped with a right-end boundary condition, can determine an implicit potential function for computing the period by an exact integral formula. However, the integrand is singular, which renders a less accurate value of the period. A generalized integral conservation law endowed with a weight function is constructed, which is proved to be equivalent to the exact integral formula. Minimizing the error to satisfy the periodicity conditions, the optimal initial value of the weight function is determined. Two non-iterative methods are developed by integrating three first-order ODEs or two first-order ODEs to compute the period. Very accurate value of the period can be observed upon testing five examples. For the NCNO without having the first integral, the integral-type period formula is derived. Four examples belong to the Liénard equation, involving the van der Pol equation, are evaluated by the proposed iterative method to determine the oscillatory amplitude and period. For the case with one or more limit cycles, the amplitude and period can be estimated very accurately. For the NCNO of a broad type with or without having the first integral, the present paper features a solid theoretical foundation and contributes integral-type formulations for the determination of the oscillatory period. The development of new numerical algorithms and extensive validation across a diverse set of examples is given. Full article

A class of boundary value problems for fractional differential systems involving variable-order derivatives is considered. Such problems can be transformed into some boundary value problems for nonlinear Caputo fractional differential systems. Here, the relations between linear Caputo fractional differential equations and their corresponding linear integral equations are investigated, and the results demonstrate that a proper Lipschitz-type condition is needed for studying nonlinear Caputo fractional differential equations. Then, an existence and uniqueness result is established in some vector subspaces by Banach’s fixed-point theorem and ${\parallel ·\parallel }_e$ norm. In addition, two examples are presented to illustrate the theoretical conclusions. Full article

A potential flow theory-based fully nonlinear 2D NWT is developed in the time domain to investigate the hydrodynamic performance of a submerged circular cylindrical WEC device under combined wave–current conditions. The hydrodynamic force on the submerged cylinder is evaluated using the acceleration potential method coupled with the desingularized boundary integral equation method (DBIEM). The impacts of the wave height, current speed, and parameters of the power take-off mechanism on the extracted power capability of the WEC device are investigated. The results show that for the scenario of an opposing current, the dimensionless mean extracted power is reduced by as much as 14.3% with increasing wave height. Except for long waves, the extracted power under a co-flowing current exceeds that of the current-free case and an opposing current yields lower power. In contrast to the current-free scenario, the peak power extraction point shifts to slightly higher values of the spring and damper constants when the current is co-flowing, whereas the opposite trend is observed for the opposing current. Full article

This study develops a coupled Smoothed Particle Hydrodynamics (SPH) and the Discrete Element Method (DEM) framework to explore the sedimentation behavior of densely arranged particles in vertical pipes. An unresolved SPH-DEM model is proposed, which integrates porosity-dependent fluid governing equations through local averaging techniques to connect pore-scale interactions with macroscopic flow characteristics. Validated against single-particle settling experiments, the model accurately captures transient acceleration, drag equilibrium, and rebound dynamics. Systematic simulations reveal that particle number, arrangement patterns, and fluid domain geometry play critical roles in regulating collective settling: Increasing particle count induces nonlinear terminal velocity reduction. Systems of 16 particles show 50% lower velocity than single-particle cases due to enhanced shielding and energy dissipation. Particle configuration (compact layouts 4 × 8 vs. elongated arrangements 8 × 4) dictates hydrodynamic resistance, compact layouts facilitate faster settling by reducing cross-sectional blockage, while elongated arrangements amplify lateral resistance. The width of the fluid domain exerts threshold effects: narrow boundaries (0.03 m) intensify wall-induced drag and suppress vortices, whereas wider domains promote symmetric vortices that enhance stability. Additionally, critical transitions in multi-row/column systems are identified, where stress-chain redistribution and fluid-permeation thresholds govern particle detachment and velocity stratification. These findings deepen the understanding of granular–fluid interactions in confined spaces and provide a predictive tool for optimizing particle management in industrial processes such as wellbore cleaning and hydraulic fracturing. Full article

This study investigates the nonlinear dynamics and control of an underactuated hybrid system consisting of a Duffing oscillator, a pendulum, and a piezoelectric energy harvester. A novel Piezoelectric Harvester Proportional–Derivative (PHPD) control scheme is introduced, which integrates the harvester’s electrical output directly into the feedback loop to achieve simultaneous vibration suppression and energy utilization. The nonlinear governing equations are derived and analyzed using the Multiple-Scale Perturbation Technique (MSPT) to obtain reduced-order dynamics. Bifurcation analysis is employed to identify stability boundaries and critical parameter transitions, while numerical simulations based on the fourth-order Runge–Kutta method validate the analytical predictions. Furthermore, frequency response curves (FRCs) and an ideal system are evaluated under multiple controller and system parameter configurations. Bifurcation classification is performed on the analyzed figure to detect various bifurcations within the system, along with the computation of the Largest Lyapunov Exponent (LLE). The results demonstrate that PHPD control significantly reduces vibration amplitude and accelerates convergence, offering a new pathway for energy-efficient, high-performance control in nonlinear electromechanical systems. Full article

Partial differential equations are an integral part of modern scientific development. Hyperbolic partial differential equations are encountered in many fields and have many applications—both linear and nonlinear types, with some being semilinear and quasilinear. In this paper, a family of implicit numerical schemes for solving hyperbolic partial differential equations is derived, utilizing finite differences and tridiagonal sweep. Through the discrete Fourier transform, a necessary and sufficient condition for convergence is proven for the linear version of the family of difference schemes, expanding the known results on boundary conditions that ensure convergence. Numerical verification confirms the found condition. A series of experiments on different boundary conditions and semilinear hyperbolic PDEs show that the same condition seems to also hold in those cases. In view of the results, an optimal subset of the family is found. A comparison between the implicit schemes and an explicit analogue is conducted, demonstrating the gained efficiency of the implicit schemes. Full article

A physics-informed neural network (PINN) framework is developed to model the large deformation and coupled electromechanical response of dielectric elastomer tubes for energy harvesting. The system integrates incompressible neo-Hookean elasticity with radial electric loading and compressible gas inflation, leading to nonlinear equilibrium equations with deformation-dependent boundary conditions. By embedding the governing equations and boundary conditions directly into its loss function, the PINN enables accurate, mesh-free solutions without requiring labeled data. It captures realistic pressure–volume interactions that are difficult to address analytically or through conventional numerical methods. The results show that internal volume increases by over 290% during inflation at higher reference pressures, with residual stretch after deflation reaching 9.6 times the undeformed volume. The axial force, initially tensile, becomes compressive at high voltages and pressures due to electromechanical loading and geometric constraints. Harvested energy increases strongly with pressure, while voltage contributes meaningfully only beyond a critical threshold. To ensure stable training across coupled stages, the network is optimized using the Optuna algorithm. Overall, the proposed framework offers a robust and flexible tool for predictive modeling and design of soft energy harvesters. Full article

This research paper delves into the application of optimality results in orthogonal fuzzy metric spaces to demonstrate the existence and uniqueness of solutions of nonlinear differential equations with boundary conditions and nonlinear integral equations, emphasizing the importance of orthogonal fuzzy metric spaces in extending fixed-point theory. Through introducing this innovative concept, the study provides a theoretical framework for analyzing mappings in diverse scenarios. In this study, we introduce the concept of best proximity point (BPP) within the framework of orthogonal fuzzy metric spaces by employing orthogonal fuzzy proximal contractive mappings. Moreover, this research explores the implications of the established results, considering both self-mappings and non-self mappings that share the same parameter set. Additionally, some examples are provided to illustrate the practical relevance of the proven results and consequences in various mathematical contexts. The findings of this study can open up avenues for further exploration and application in solving real-world problems. Full article

A method for selecting the optimal shape of holes, taking into account the strength anisotropy of composites, is proposed. The methodology includes the following: an algorithm for stress determination based on singular integral equations and Green’s solutions; a strength criterion for the boundary of unloaded holes, which takes into account the anisotropic mechanical and strength properties of composites; an algorithm for determining hole shapes by a formulated nonlinear programming problem. The results of the research are presented for holes of various shapes, including single- and double-periodic hole systems. It is established that the calculated allowable loads for composite plates with holes based on stress concentration factors can be significantly overestimated. At the same time, by designing holes of optimal shape, the allowable loads can be many times greater than those for circular holes. Full article

The objective of this work is to investigate the global existence, general decay and blow-up results for a class of p-Biharmonic-type hyperbolic equations with delay and acoustic boundary conditions. The global existence of solutions has been obtained by potential well theory and the general decay result of energy has been established, in which the exponential decay and polynomial decay are only special cases, by using the multiplier techniques combined with a nonlinear integral inequality given by Komornik. Finally, the blow-up of solutions is established with positive initial energy. To our knowledge, the global existence, general decay, and blow-up result of solutions to p-Biharmonic-type hyperbolic equations with delay and acoustic boundary conditions has not been studied. Full article

In this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring variable coefficients and a functional boundary condition. To demonstrate the applicability of our main theorems, we provide several examples in which we compute values of the two-parameter Mittag–Leffler functions. The proposed approach is particularly effective for addressing a wide range of integral and fractional nonlinear differential equations with initial or boundary conditions—especially those involving variable coefficients, which are typically challenging to treat using classical integral transform methods. Finally, we demonstrate a significant application of the inverse operator approach by solving a Caputo fractional convection partial differential equation in ${\mathbb{R}}^n$ with an initial condition. Full article

This study proposes an intelligent prediction framework integrating native sparse attention (NSA) with the Chen-Guan (CHG) algorithm to optimize tunnel boring machine (TBM) operations in heterogeneous geological environments. The framework resolves critical limitations of conventional experience-driven approaches that inadequately address the nonlinear coupling between the spatial heterogeneity of rock mass parameters and mechanical system responses. Three principal innovations are introduced: (1) a hardware-compatible sparse attention architecture achieving O(n) computational complexity while preserving high-fidelity geological feature extraction capabilities; (2) an adaptive kernel function optimization mechanism that reduces confidence interval width by 41.3% through synergistic integration of boundary likelihood-driven kernel selection with Chebyshev inequality-based posterior estimation; and (3) a physics-enhanced modelling methodology combining non-Hertzian contact mechanics with eddy field evolution equations. Validation experiments employing field data from the Pujiang Town Plot 125-2 Tunnel Project demonstrated superior performance metrics, including 92.4% ± 1.8% warning accuracy for fractured zones, ≤28 ms optimization response time, and ≤4.7% relative error in energy dissipation analysis. Comparative analysis revealed a 32.7% reduction in root mean square error (p < 0.01) and 4.8-fold inference speed acceleration relative to conventional methods, establishing a novel data–physics fusion paradigm for TBM control with substantial implications for intelligent tunnelling in complex geological formations. Full article

Geometric image transformations are fundamental to image processing, computer vision and graphics, with critical applications to pattern recognition and facial identification. The splitting-integrating method (SIM) is well suited to the inverse transformation $T^{-1}$ of digital images and patterns, but it encounters difficulties in nonlinear solutions for the forward transformation T. We propose improved techniques that entirely bypass nonlinear solutions for T, simplify numerical algorithms and reduce computational costs. Another significant advantage is the greater flexibility for general and complicated transformations T. In this paper, we apply the improved techniques to the harmonic, Poisson and blending models, which transform the original shapes of images and patterns into arbitrary target shapes. These models are, essentially, the Dirichlet boundary value problems of elliptic equations. In this paper, we choose the simple finite difference method (FDM) to seek their approximate transformations. We focus significantly on analyzing errors of image greyness. Under the improved techniques, we derive the greyness errors of images under T. We obtain the optimal convergence rates $O\left(H^2\right)+O(H/N^2)$ for the piecewise bilinear interpolations ($\mu =1$) and smooth images, where $H(\ll 1)$ denotes the mesh resolution of an optical scanner, and N is the division number of a pixel split into $N^2$ sub-pixels. Beyond smooth images, we address practical challenges posed by discontinuous images. We also derive the error bounds $O\left(H^{\beta }\right)+O(H^{\beta }/N^2)$, $\beta \in (0,1)$ as $\mu =1$. For piecewise continuous images with interior and exterior greyness jumps, we have $O\left(\sqrt{H}\right)+O(\sqrt{H}/N^2)$. Compared with the error analysis in our previous study, where the image greyness is often assumed to be smooth enough, this error analysis is significant for geometric image transformations. Hence, the improved algorithms supported by rigorous error analysis of image greyness may enhance their wide applications in pattern recognition, facial identification and artificial intelligence (AI). Full article