Tunnel lighting cleaning is of significant practical importance for improving driving safety. To address the low operational efficiency of tunnel lighting cleaning tasks, a trajectory planning method based on the chaotic adaptive whale-particle swarm optimization (CAW-PSO) algorithm is proposed. Taking the SIASUN GCR16-2000 robotic arm as the research object, the trajectory is constructed using a 3-5-3 polynomial interpolation, with the objective of achieving time-optimal trajectory planning. In the CAW-PSO algorithm, a tent chaotic map is introduced to improve the quality of the population; a linearly decreasing inertia weight is designed to strike a balance between local and global search; dynamic learning factors are defined to strengthen the individual learning ability and global cognitive capability of particles; finally, the exploitation mechanism of the whale optimization algorithm is employed to avoid getting trapped in local optima and improve convergence accuracy. The simulation time is 3.661 s, a reduction of 69.94%. The experimental results yielded a mean relative error of 1.16%, indicating good agreement with the simulation results. The results of the simulation and experiment indicate that the CAW-PSO effectively reduces the motion time of the robotic arm, exhibiting superior applicability in trajectory planning for tunnel lighting cleaning robotic arms.

A high-order finite element method based on Hermite interpolation polynomials is described and a software implementation in the deal.II finite element library (https://www.dealii.org) is proposed. This method can be used to strongly enforce continuity of spatial derivatives as well as solution value over element boundaries, which is necessary for solving fourth order partial differential equations (PDEs) in primal form and useful for some second order PDEs. The implementation is verified by solving the wave equation based on a modification of the PDE with a Lax-Wendroff-like procedure, allowing a high-order solution method with negligible additional computing time.

Global Solar Radiation (Rs) is essential for ecological and climatic modeling, yet its spatialization is often hampered by sparse observation networks. Conventional methods demand a well-distributed set of stations with global representativeness—a requirement rarely met in practice. To address this gap, we propose a spatialization method based on environmental similarity and spatial proximity (ES-SP), which integrates the Law of Geographic Similarity and Tobler’s First Law of Geography. Using monthly Rs data from 11 stations in Tropical China (2015), we evaluated ES-SP against Ordinary Kriging (OK) and Local Polynomial Interpolation (LP) through leave-one-out cross-validation (LOOCV), with root mean square error (RMSE), relative RMSE, and mean absolute percentage error (MAPE) as accuracy metrics. Topographic and monthly meteorological covariates were selected dynamically via random forest (RF), and the performance differences among the three methods were tested statistically using the Wilcoxon signed-rank test. Results show that ES-SP outperforms both OK and LP in accuracy and stability, achieving the lowest error metrics in most months—e.g., RMSE as low as 37.23 MJ·m−2 in December and MAPE as low as 4.34% in August—along with a narrow interquartile range, indicating consistent performance across seasons. Spatially, ES-SP accurately reproduces the coastal–inland gradient during the rainy season (May) and the latitudinal gradient in the dry season (January), whereas OK yields overly smooth distributions that obscure local details, and LP exhibits extreme instability and unrealistic spatial discontinuities. The study demonstrates that the ES-SP method effectively overcomes the reliance on globally representative station samples, providing a robust technical pathway for generating continuous Rs datasets in data-sparse regions such as Tropical China. Further research should focus on extending the geographic scope and refining the covariate set to enhance generalizability.

A multiple surrogate simulation-optimization framework for designing pump-and-treat systems.

We develop a novel Steffensen-type iterative solver to solve nonlinear scalar equations without requiring derivatives. A two-parameter one-step scheme without memory is first introduced and analyzed. Its optimal quadratic convergence is then established. To enhance the convergence rate without additional functional evaluations, we extend the scheme by incorporating memory through adaptively updated accelerator parameters. These parameters are approximated by Newton interpolation polynomials constructed from previously computed values, yielding a derivative-free method with R-rate of convergence of approximately 3.56155. A dynamical system analysis based on attraction basins demonstrates enlarged convergence regions compared to Steffensen-type methods without memory. Numerical experiments further confirm the accuracy of the proposed scheme for solving nonlinear equations.

It is known that every function with a finite support over a given field can be interpolated by means of the Lagrangian polynomial. The question is if a similar interpolation is possible if one considers a unitary ring or a Boolean algebra instead of a field. We get a positive answer to this question provided the similarity type of the algebra in question is enriched with one more unary operation, the so-called Baaz delta. We get an explicit construction of this interpolation polynomial in both the cases. When going to Boolean posets, we have a lack of operations but these can be substituted by the operators [Formula: see text] and [Formula: see text]. Hence, we generalize also the Baaz delta for posets as an operator and then we can derive an explicit interpolation term constructed by means of these operators also for Boolean posets.

In this study, we formulate a deterministic mathematical model to describe the transmission dynamics of the monkeypox virus using fractal and fractional-order differential equations. The model incorporates all possible interactions influencing disease propagation within the population. Our analysis primarily focuses on the stability of fractal–fractional derivatives, aiming to establish the existence and uniqueness of solutions through the fixed-point theorem. Additionally, we examine Ulam-Hyers stability and other significant findings related to the proposed model. To enhance numerical accuracy, we employ Lagrange polynomial interpolation for computational approximations. Finally, graphical simulations for various fractal–fractional orders are presented to validate the model’s effectiveness and demonstrate its practical relevance.

In computational applications, it is often the case that measurements are collected at uniformly distributed locations. In such cases, using ordinary polynomial interpolation may lead to divergence due to the Runge phenomenon; furthermore, the interpolation process is known to be severely ill-conditioned. To address these challenges, an effective strategy involves selecting mock-Chebyshev points for polynomial interpolation from a dense set of uniformly spaced points, thereby replicating the favorable properties of Chebyshev nodes. Yet, few studies in the literature address the computation of these nodes. Moreover, developing an algorithm that can generate mock-Chebyshev points with O ( n ) computational complexity is of significant practical importance. This study proposes a new version of the fast algorithm introduced in Ibrahimoglu [A fast algorithm for computing the mock-Chebyshev nodes, J. Comput. Appl. Math. 373 (2020), p. 112336.], which employs the floor function to calculate the ratio of distances between consecutive Chebyshev–Lobatto interpolation points. The resulting algorithm is shown to be fast and stable, consistently generating a distribution of nodes fulfilling the mock-Chebyshev requirements with the linear complexity O ( n ) , simultaneously reducing the cardinality of the corresponding satisfactory uniform grid. This study also introduces a theoretical lower bound on the minimum number of equispaced nodes required to satisfy the mock-Chebyshev conditions. A bivariate extension of the approach to mock-Padua points on [ − 1 , 1 ] 2 is also presented and validated through numerical experiments.

A surrogate for computationally efficient crystal plasticity modeling via database and polynomial interpolation

Strictly positive definite functions of finite orders and multivariate polynomial interpolation

This paper focuses on providing an empirical model construction methodology based on Piecewise Cubic Hermite Interpolation Polynomials (PCHIP). The proposed methodology is suitable for modelling systems with high non-linearity between system parameters, process variables and/or operation variables. A detailed description of the 4D interpolation methodology is given and a practical application for a fuel cell (FC) blower black-box model obtaining is shown. The results of the algorithm-based model are compared to datasheet specifications to validate the FC blower model. The validation shows a good precision in the operation limits estimation. Additionally, some FC-related applications suitable for the obtained blower model utilization are described. Key words. Piecewise Cubic Hermite Interpolation Polynomials (PCHIP), Modelling, Fuel Cells (FC), Blower.

This research presents a novel single-step hybrid block method with seven intra-step points that achieves ninth-order accuracy, providing an accurate and computationally efficient approach for solving first-order stiff differential equations. The method is designed to solve first-order stiff differential equations with high efficiency and precision while maintaining a constant step size throughout the computation. To further improve accuracy, Lagrange polynomial interpolation techniques are employed to approximate function values at selected points within each step. The fundamental properties of the proposed scheme are rigorously analysed to establish its mathematical validity. These analyses confirm that the method satisfies the essential conditions of consistency, stability, and convergence, thereby ensuring its reliability for applications. The proposed method performs effectively when applied to stiff and oscillatory differential equations. Comprehensive numerical experiments are conducted, and the results consistently demonstrate the robustness and effectiveness of the proposed method across various test problems. Furthermore, the findings indicate that the method often outperforms several existing numerical techniques in terms of both accuracy and computational efficiency.

In this work, we develop a numerical framework for solving second-order fractional differential equations. Second-order fractional differential equations involving the Atangana–Baleanu Caputo fractional derivative, which models nonlocal and memory-dependent dynamical behavior. The numerical scheme is constructed using Lagrange polynomial interpolation adapted to the Atangana–Baleanu Caputo operator. A rigorous convergence and stability analysis is carried out using a fixed-point (contraction) argument under a natural Lipschitz condition. The theoretical results are supported by several numerical experiments that cover linear and nonlinear test problems. The contribution of this study lies in the establishment of a stable and accurate ABC-based numerical scheme and the verification that it standardizes its performance across a wide range of source functions. In general, the proposed method appears to be a powerful and efficient tool for modeling and analyzing complex systems that obey fractional dynamics.

Curve reconstruction is the process of estimating a smooth function or curve that fits a given setof data points, either exactly (interpolation) or approximately (fitting). Classical approaches,including global polynomial interpolation, splines, Hermite interpolation, and radial basisfunction fitting, face challenges when data are sparse, irregularly distributed, or noisy. In thispaper, we propose a curve reconstruction method based on the discrete form of the biharmonicequation. The method formulates reconstruction as a constrained quadratic optimizationproblem, incorporating both equality and inequality constraints and producing globallyC1smooth curves. The approach is physically interpretable, penalizing excessive bending, as in thecase of a thin elastic beam, and can be extended to higher-dimensional surface reconstruction.Performanceisevaluatedthroughnumericalexperimentsonknownfunctionsandsyntheticdatawith various distributions and constraints, including small perturbation tests to assess stabilityand robustness. The results demonstrate that the proposed method reproduces the data, enforcesthe prescribed bounds, and remains stable under irregular sampling and noise.

Quadrature formulas are constructed for singular integrals on the integration segment [-1, 1], with weight functions (t)=1/√(1-t^2 ), p(t)=√(1-t^2 ). The construction uses the Lagrange interpolation polynomial with function values at fixed points {-1; 1} and at zeros of Chebyshev polynomials orthogonal to [-1, 1] with respect to the corresponding weight functions. When calculating the coefficients of the quadrature formula, formulas for singular integrals are used, where Chebyshev polynomials of the first and second genera are taken as the density. The formulas obtained are quadrature formulas of the interpolation type. The paper provides examples confirming the effectiveness of the obtained quadrature formulas for calculating singular integrals. In frequency, the functions φ(t)=1,φ(t)=t and φ(t)=t^2 are used as density, for which accurate results are obtained. Estimates of the error are given for the obtained quadrature formulas. The resulting quadrature formulas have an algebraic degree of accuracy of n+1.

New bounds for identities related to generalizations of Steffensen’s inequality via Abel-Gontscharoff’s and Hermite’s interpolation polynomials

The properties of films assembled by drying colloidal-particle suspensions depend sensitively on both the particles and the processing conditions, making them challenging to engineer. In this work, we develop and test an inverse-design strategy based on surrogate modeling to identify conditions that yield a target film structure. We consider a two-component hard-sphere colloidal suspension whose designable parameters are the particle sizes, the initial composition of particles, and the drying rate. Film drying is simulated approximately using Brownian dynamics. Surrogate models based on Gaussian process regression (GPR) and Chebyshev polynomial interpolation are trained on a loss function, computed from the simulated film structures, that guides the design process. We find the surrogate models to be effective for both approximation and optimization using only a small number of samples of the loss function. The GPR models are typically slightly more accurate than polynomial interpolants trained using comparable amounts of data, but the polynomial interpolants are more computationally convenient. This work has important implications not only for designing colloidal materials but also more broadly as a strategy for engineering nonequilibrium assembly processes.

Before deploying algorithms in industrial settings, it is essential to validate them in virtual environments to anticipate real-world performance, identify potential limitations, and guide necessary optimizations. This study presents the development and integration of artificial intelligence algorithms for detecting labels and container formats of cleaning products using computer vision, enabling robotic manipulation via a UR5 arm. Label identification is performed using the speeded-up robust features (SURF) algorithm, ensuring robustness to scale and orientation changes. For container recognition, multiple methods were explored: edge detection using Sobel and Canny filters, Hopfield networks trained on filtered images, 2D cross-correlation, and finally, a you only look once (YOLO) deep learning model. Among these, the custom-trained YOLO detector provided the highest accuracy. For robotic control, smooth joint trajectories were computed using polynomial interpolation, allowing the UR5 robot to execute pick-and-place operations. The entire process was validated in the CoppeliaSim simulation environment, where the robot successfully identified, classified, and manipulated products, demonstrating the feasibility of the proposed pipeline for future applications in semi-structured industrial contexts.

Abstract The interface problem arising from the dendritic solidification of pure substances is governed by the distribution of heat in a phase‐changing medium. This study develops an efficient layer‐adapted difference scheme for a mathematical interface model formulated as a time‐fractional convection‐diffusion problem featuring a discontinuous convection coefficient. The fractional derivative of order is interpreted in the Caputo sense. The computational domain is divided into two subdomains, each governed by a time‐fractional convection‐diffusion equation. Due to the mild singularity at , the solution typically exhibits an initial layer, which adversely affects the accuracy of standard polynomial interpolation on uniform meshes. To overcome this challenge and ensure optimal convergence, we employ the L1 discretization for the fractional derivative on a nonuniform graded mesh. The spatial derivatives are approximated using a second‐order finite difference scheme. The convergence analysis is carried out using the discrete comparison principle along with a suitably chosen barrier function, leading to rigorous error estimates. Numerical experiments are provided to support the theoretical results, confirming the accuracy, efficiency, and robustness of the proposed scheme.

In this paper, a new class of fifth-order Chebyshev–Halley-type methods with a single parameter is proposed by using the polynomial interpolation method. The convergence order of the new method is proved. The dynamic behavior of the new method on quadratic polynomials P(x)=(x−a)(x−b) is analyzed, the strange fixed points and the critical points of the operator are obtained, the corresponding parameter planes and dynamic planes are drawn, the stability and convergence of the iterative method are visualized, and some parameter values with good properties are selected. The fractal results of the new method corresponding to different parameters about polynomial G(x) are plotted. Numerical results show that the new method has less computing and higher computational accuracy than the existing Chebyshev–Halley-type methods. The fractal results show the new method has good stability and convergence. The numerical results of different iteration methods are compared and agree with the results of dynamic analysis.