Search Results (1,839)

Despite the significant interest from research communities in understanding the squeezing flow of fluid between two plates, important qualitative and quantitative questions regarding solutions to these squeezing flow models still remain unanswered, including existence, uniqueness, location, approximation, and convergence. Thus, the purpose of the present paper is to construct a firm mathematical basis that establishes the above knowledge for the squeezing flow model and its boundary value problem. Full article

In this paper, we investigate a class of nonlinear fractional boundary value problems involving the Caputo fractional derivative under two-point boundary conditions. By combining the Green function of the associated linear problem with a generalized Gronwall inequality, we derive pointwise estimates for solutions expressed explicitly in terms of the Mittag–Leffler function. In contrast to existing Lyapunov-type inequalities, which are mainly restricted to linear equations and rely on global supremum norm estimates, our approach preserves the nonlinear structure of the problem and captures the local behavior of solutions. These pointwise estimates lead to a Lyapunov-type inequality for nonlinear fractional equations, extending the classical result of Jleli and Samet beyond the linear framework. Moreover, we show that the obtained Lyapunov condition serves not only as a necessary condition for the existence of nontrivial solutions, but also as a sufficient criterion ensuring Hyers–Ulam stability and uniqueness. An illustrative example is provided to demonstrate the applicability of the theoretical results. Full article

This study employs the Atangana–Baleanu–Caputo fractional derivative within the Moore–Gibson–Thompson heat conduction model to analytically investigate the thermoelastic vibrations in solid medium-containing voids. The ABC–MGT formulation incorporates a non-singular Mittag–Leffler memory kernel, facilitating the modeling of tempered hereditary relaxation in voided thermoelastic media, thereby producing more realistic attenuation and phase lag characteristics in transient responses than conventional integer-order models. Specifically, our novelty lies in developing a coupled thermoelastic–void formulation within an ABC–MGT heat conduction framework, deriving the full governing system and boundary-value solution in the Laplace domain, and providing a systematic parametric analysis showing how the ABC order changes attenuation, phase lag, and stress/void interactions. This approach enables a precise analytical resolution of the problem. The analysis indicates that the presence and size of voids substantially impact the system response variables, with smaller apertures yielding reduced magnitudes. Thus, this analytical investigation introduces a novel methodology for addressing the complex challenges associated with advanced functional materials and high-performance engineering structures. Full article

Path planning for large-scale agricultural fields faces challenges such as irregular field shapes, uncertain boundaries, and the need to balance path efficiency, energy consumption, and coverage quality. To address these problems, this research introduces a strategy-aware hierarchical hybrid optimization framework (HANS) for autonomous agricultural operations. This framework introduces a global principal axis extraction method based on Principal Component Analysis (PCA), utilizing the statistical distribution of field boundaries to guide path direction, thereby improving robustness against boundary noise and irregular geometries. The framework integrates Adaptive Large Neighborhood Search (ALNS) for global exploration and Tabu Search (TS) for local optimization, forming a tightly coordinated hybrid structure. The framework further employs a Pareto-set-based decision support selection strategy to solve a multi-objective optimization model encompassing machine kinematics, turning patterns, and energy-aware cost evaluation. This strategy provides three methods: weighted preference-based compromise solution selection, crowding distance-based diversified solution selection, and single-objective extreme value-based dedicated optimization solution selection. To balance the impact of path length, energy consumption, and coverage rate, we assigned equal or nearly equal weights to them (i.e., (0.33, 0.33, 0.34)). Furthermore, the framework incorporates operators and feedback learning mechanisms specific to agricultural coverage path problems to enable adaptive operator selection and reduce reliance on manual parameter tuning. Simulation results under three representative field scenarios show that compared to fixed-direction planning, HANS improves the average coverage rate by 0.51 percentage points and reduces fuel consumption by 4.34%. Compared to Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Tabu Search (TS), and Simulated Annealing (SA), the proposed method shortens the working path length by 0.37–0.83%, improves coverage rate by 0.34–1.11%, and reduces energy consumption by 0.61–1.03%, while maintaining competitive computational costs. These results demonstrate the effectiveness and practicality of HANS in large-scale autonomous farming operations. Full article

We propose a two-level theory that connects Lin-equation-based dynamical coarse-graining of the turbulence cascade with an information-theoretic selection principle in logarithmic wavenumber space. This framework places the dissipation-range spectral shape on a verifiable logical basis rather than on ad hoc fitting. At the first (dynamical) level, we formulate an autonomous conservative Fokker–Planck equation for the normalized density and probability current. Under sufficient boundary decay and a strictly positive effective diffusion, the sign-reversed Kullback–Leibler divergence is shown to be a Lyapunov functional, yielding a rigorous H-theorem and fixing the arrow of time in scale space. At the second (selection) level, the dissipation range is treated as a stationary boundary-value problem for an open system by introducing a killing term for an unnormalized scale density. A WKB (Liouville–Green) analysis restricts the admissible tail to a stretched-exponential form and links the tail exponent to the high-wavenumber scaling of the effective diffusion. The exponential prefactor is fixed by dissipation-rate consistency, and the remaining degree of freedom is determined by one-dimensional Kullback–Leibler minimization (Hyper-MaxEnt) against a globally constructed reference distribution. The resulting exponent range is validated against the high-resolution DNS spectra reported in the literature. Full article

This paper introduces polynomial F-contractions, a novel category of contractive mappings within metric spaces. This concept synthesizes two powerful generalizations of the Banach contraction principle: the F-contractions originally developed by Wardowski and the polynomial-type contractions studied very recently by Jleli et al. We formulate fixed point theorems for this new class of mappings in complete metric spaces, which extends and unifies several established theorems in fixed point theory. We first prove our main result for continuous mappings and then extend it to a broader class of mappings that are not necessarily continuous but satisfy the Picard continuity condition. The significance and novelty of our results are highlighted through illustrative examples and further supported by applications to a fractional boundary value problem. Full article

This paper studies mixed impulsive boundary value problems involving tempered $\mathsf{\Psi }$-fractional derivatives of Caputo type. By introducing exponential tempering into the fractional framework, the proposed model effectively captures systems with fading memory—an improvement over conventional power-law kernels that assume long-range dependence. The generalized tempered $\mathsf{\Psi }$-operator unifies several existing fractional derivatives, offering enhanced flexibility for modeling complex dynamical phenomena. Impulsive effects and integral boundary conditions are incorporated to describe processes subject to sudden changes and historical dependence. The problem is reformulated as a Volterra integral equation, and fixed-point theory is employed to establish analytical results. Existence and uniqueness of solutions are proven using the Banach Contraction Mapping Principle, while the Leray–Schauder nonlinear alternative ensures existence in non-contractive cases. The proposed framework provides a rigorous analytical basis for modeling phenomena characterized by both fading memory and sudden perturbations, with potential applications in physics, control theory, population dynamics, and epidemiology. A numerical example is presented to illustrate the validity and applicability of the main theoretical results. Full article

Boundary value problems for systems of integrodifferential equations appear in many branches of science and engineering. Accuracy in modeling complex processes requires the specification of nonlocal boundary conditions, including multipoint and integral conditions. These kinds of problems are even harder to solve. In this paper, we present solvability criteria and a direct operator method for constructing the exact solution to systems of linear ordinary integrodifferential equations with general nonlocal boundary conditions. A symbolic algorithm is also proposed. Several examples are solved to demonstrate the effectiveness of the method. The results obtained are equally valid for nonlocal boundary value problems for systems of ordinary differential, loaded differential, and loaded integrodifferential equations. Full article

With the advancement of the dual carbon goals and the rapid increase in the proportion of new energy installations, the power system faces multiple challenges including insufficient flexibility resources, intensified fluctuations in generation and load, and reduced operational safety. Integrated energy systems (IESs), serving as key platforms for integrating diverse energy sources and flexible resources, possess complex internal structures and limited individual regulation capabilities, making direct participation in grid dispatch and market interactions challenging. To achieve large-scale resource coordination and efficient utilization, this paper investigates external characteristic modeling and cluster aggregation optimization methods for IES, proposing a comprehensive technical framework spanning from individual external characteristic identification to cluster-level coordinated control. First, addressing the challenge of unified dispatch for heterogeneous resources within IES, this study proposes an external characteristic modeling method based on operational feasible region projection. It constructs models for the active power output boundary, marginal cost characteristics, and ramping rate of virtual power plants (VPPs), enabling quantitative representation of their overall regulation potential. Second, a cluster aggregation optimization model for integrated energy systems is established, incorporating regional autonomy. This model pursues multiple objectives: cost–benefit matching, maximizing renewable energy absorption rates, and minimizing peak external power purchases. The Gini coefficient and Shapley value method are introduced to ensure fairness and participation willingness among cluster members. Furthermore, an optimization mechanism incorporating key constraints such as cluster scale, grid interaction, and regulation complementarity is designed. The NSGA-II multi-objective genetic algorithm is employed to efficiently solve this high-dimensional nonlinear problem. Finally, simulation validation is conducted on a typical regional energy scenario based on the IEEE-57 node system. Results demonstrate that the proposed method achieves average daily cost savings of approximately 3955 CNY under the optimal aggregation scheme, reduces wind and solar curtailment rates to 5.38%, controls peak external power purchases within 2292 kW, and effectively incentivizes all entities to participate in coordinated regulation through a rational benefit distribution mechanism. Full article

Aiming at the key problems of the material removal rate and surface integrity of existing tools in the lapping of sapphire hard and brittle crystals, an efficient lapping tool has been developed to explore a new process for HFVCD (hot filament chemical vapor deposition) diamond tools to efficiently lap sapphire wafers. With the premise of ensuring the surface roughness of the wafer is Ra ≤ 0.5 μm, the material removal rate is increased to more than 1 μm/h. To explore a high-efficiency lapping process for sapphire wafers using HFCVD diamond tools. The influence of key preparation parameters on the surface characteristics of CVD (chemical vapor deposition) diamond films was systematically investigated. Three types of CVD diamond coating tools with distinct surface morphologies were fabricated. These tools were subsequently employed to conduct lapping experiments on sapphire wafers in order to evaluate their processing performance. The experimental results demonstrate that the gas pressure, methane concentration, and substrate temperature collectively influenced the surface morphology of the diamond coatings. The fabricated coatings exhibited well-defined grain boundaries and displayed pyramidal, prismatic and spherical features, corresponding to high-quality microcrystalline and nanocrystalline diamond layers. In the lapping experiments, the prismatic CVD diamond coating tool exhibited the highest material removal rate, reaching approximately 1.7 μm/min once stabilized. The spherical diamond coating tool produced the lowest surface roughness on the lapped sapphire wafers, with a value of about 0.35 μm. Surface morphology-controllable diamond tools were used for the lapping processing of the sapphire wafers. This achieved a good surface quality and high removal rate and provided new ideas for the precision machining of brittle hard materials in the plane or even in the curved surface. Full article

This study introduces a mathematical framework for analyzing unsteady gas filtration in porous media with pressure-dependent porosity variations. The physical process is formulated as a nonlinear parabolic boundary value problem that captures the coupled interaction between pressure evolution and porosity changes during gas production. To solve the equation, a numerical strategy is developed by integrating the Alternating Direction Implicit (ADI) scheme with quasi-linearization iterations, employing finite difference discretization on a two-dimensional spatial grid. Extensive computational experiments are performed to investigate the influence of key reservoir parameters—including porosity coefficient, permeability, gas viscosity, and well production rate—on the spatiotemporal behavior of pressure and porosity during long-term extraction. The results indicate significant porosity variations near the wellbore driven by local pressure depletion, reflecting strong sensitivity of the system to formation properties. The validated numerical model provides valuable quantitative insights for optimizing reservoir management and improving production forecasting in gas field development. Overall, the proposed methodology serves as a practical tool for oil and gas engineers to assess long-term reservoir performance under diverse operational conditions and to design efficient extraction strategies that incorporate pressure-dependent formation property changes. Full article

The maximum principle is a widely used qualitative property of linear (and not only) elliptic boundary value problems. A natural goal for developing numerical methods is for the approximate solution to have a similar property. In this case, we say that a discrete maximum principle holds. In many cases, such a requirement is critical to ensuring the reliability of computational models. Here, we consider multidimensional linear elliptic problems with diffusion and reaction terms. Such problems have been studied and analyzed for many decades. Since relatively recently, scientists have faced conceptually new challenges when considering anomalous (fractional) diffusion. In the present paper, we concentrate on the case of spectral fractional diffusion. Discretization was carried out using the finite difference method and the finite element method with a lumped mass matrix. In large-scale multidimensional problems, the computational complexity of dense matrix operations is critical. To overcome this problem, BURA (best uniform rational approximation) methods were applied to find the efficient numerical solutions of emerging dense linear systems. Thus, along with the need to satisfy the discrete maximum principle associated with the mesh method applied for discretization of the differential operator, the issue of the monotonicity of BURA numerical solution arises. The presented results are three-fold and include the following: (i) maximum principles for fractional diffusion–reaction problems; (ii) sufficient conditions for discrete maximum principles; and (iii) sufficient conditions for monotonicity of the investigated BURA- or BURA-like approximation methods. A novel, systematic theoretical analysis is developed for sub-diffusion with a fractional power $\alpha \in (1/2,1)$ and a constant reaction coefficient. The theoretical findings are further supported by numerical examples. Full article

This paper proposes a method based on interval linear robust optimization to address the potential impacts of multiple uncertainties on the operational security of Regional Integrated Energy Systems (RIESs). The model considers the uncertainty in user loads and renewable energy outputs and determines the value ranges of related parameters through statistical analysis to characterize the boundaries of these uncertainties. To transform the stochastic disturbances into a solvable problem, the model introduces energy balance constraints under the worst-case scenario, ensuring that the system remains feasible under extreme conditions. The research framework integrates Nash bargaining theory, demand response mechanisms, and tiered carbon trading policies, constructing a cooperative game model for RIESs to minimize the overall operation cost of the alliance while providing a reasonable revenue distribution scheme. This approach aims to achieve fairness and sustainability in regional cooperation. Simulation results show that the method can effectively reduce the collaborative operation cost and improve the fairness of revenue distribution. To address potential issues of information misreporting and dishonesty in real-world scenarios, the model introduces an adjustable fraud factor in the revenue distribution process to characterize the strategy deviations of participants. Even under potential fraud risks, the mechanism can maintain an optimal revenue structure and lead the participants toward a stable fraud equilibrium, thereby enhancing the robustness and reliability of the overall collaboration. Full article

This study investigates the two-dimensional (2D) steady-state frictional contact behavior of functionally graded piezoelectric material (FGPM) coatings under a high-speed rigid cylindrical punch. An electromechanical coupled contact model considering inertial effects is established, while a layered model is employed to simulate arbitrarily varying material parameters. Based on piezoelectric elasticity theory, the steady-state governing equations for the coupled system are derived. By utilizing the transfer matrix method and the Fourier integral transform, the boundary value problem is converted into a system of coupled Cauchy singular integral equations of the first and second kinds in the frequency domain. These equations are solved semi-analytically, using the least squares method combined with an iterative algorithm. Taking a power-law gradient distribution as a case study, the effects of the gradient index, relative sliding speed, and friction coefficient on the contact pressure, in-plane stress, and electric displacement are systematically analyzed. Furthermore, the contact responses of FGPM coatings with power-law, exponential, and sinusoidal gradient profiles are compared. The findings provide a theoretical foundation for the optimal design of FGPM coatings and for enhancing their operational reliability under high-speed service conditions. Full article

This research work investigates the existence, uniqueness, and stability of solution for a time-fractional fourth-order partial differential equation, subject to two initial conditions and four nonlocal integral boundary conditions. The equation incorporates several key components: the Caputo fractional derivative operator, the Laplace operator, the biharmonic operator, as well as terms representing frictional and viscoelastic damping. The presence of these elements, particularly the nonlocal boundary constraints, introduces new mathematical challenges that require the development of advanced analytical methods. To address these challenges, we construct a functional analytic framework based on Sobolev spaces and employ energy estimates to rigorously prove the well-posedness of the problem. Full article