Search Results (49)

In the theory of mixed-type equations, there are many works in bounded domains with smooth boundaries bounded by a normal curve for first- and second-kind mixed-type equations. In this paper, for a second-kind mixed-type equation in an unbounded domain whose elliptic part is a horizontal half-strip, a Bitsadze–Samarskii-type problem is investigated. The uniqueness of the solution is proved using the extremum principle, and the existence of the solution is proved by the Green’s function method and the integral equations method. When constructing the Green’s function, the properties of Bessel functions of the second kind with imaginary argument and the properties of the Gauss hypergeometric function are widely used. Visualization of the solution to the Bitsadze–Samarskii-type problem is performed, confirming its correctness from both mathematical and physical points of view. Full article

Advection–diffusion–reaction-type interface models have wide-ranging applications in environmental science, chemical engineering, and biological systems, particularly in modeling pollutant transport in groundwater, reactive flows, and drug diffusion across biological membranes. This paper presents a novel numerical method for the solution of these models. The proposed method integrates the meshless collocation technique with the finite difference method. The temporal derivative is approximated using a finite difference scheme, while spatial derivatives are approximated using radial basis functions. The interface across the fixed boundary is treated with discontinuous diffusion, advection, and reaction coefficients. The proposed numerical scheme is applied to both linear and non-linear models. The Gauss elimination method is used for the linear models, while the quasi-Newton linearization method is employed to address the non-linearity in non-linear cases. The $L_{\infty }$ error is computed for varying numbers of collocation points to assess the method’s accuracy. Furthermore, the performance of the method is compared with the Haar wavelet collocation method and the immersed interface method. Numerical results demonstrate that the proposed approach is more efficient, accurate, and easier to implement than existing methods. The technique is implemented in MATLAB R2024b software. Full article

Accurate 3D semantic occupancy perception is critical for autonomous driving, enabling robust navigation in unstructured environments. While vision-based methods suffer from depth inaccuracies and lighting sensitivity, LiDAR-based approaches face challenges due to sparse data and dependence on expensive manual annotations. This work proposes LiGaussOcc, a novel self-supervised framework for dense LiDAR-based 3D semantic occupancy prediction. Our method first encodes LiDAR point clouds into voxel features and addresses sparsity via an Empty Voxel Inpainting (EVI) module, refined by an Adaptive Feature Fusion (AFF) module. During training, a Gaussian Primitive from Voxels (GPV) module generates parameters for 3D Gaussian Splatting, enabling efficient rendering of 2D depth and semantic maps. Supervision is achieved through photometric consistency across adjacent camera views and pseudo-labels from vision–language models, eliminating manual 3D annotations. Evaluated on the nuScenes-OpenOccupancy benchmark, LiGaussOcc achieved performance competitive with 30.4% Intersection over Union (IoU) and 14.1% mean Intersection over Union (mIoU). It reached 91.6% of the performance of the fully supervised LiDAR-based L-CONet, while completely eliminating the need for costly and labor-intensive manual 3D annotations. It excelled particularly in static environmental classes, such as drivable surfaces and man-made structures. This work presents a scalable, annotation-free solution for LiDAR-based 3D semantic occupancy perception. Full article

This study presents the Fourier–Gegenbauer integral Galerkin (FGIG) method, a new numerical framework that uniquely integrates Fourier series and Gegenbauer polynomials to solve the one-dimensional advection–diffusion (AD) equation with spatially symmetric periodic boundary conditions, achieving exponential convergence and reduced computational cost compared to traditional methods. The FGIG method uniquely combines Fourier series for spatial periodicity and Gegenbauer polynomials for temporal integration within a Galerkin framework, resulting in highly accurate numerical and semi-analytical solutions. Unlike traditional approaches, this method eliminates the need for time-stepping procedures by reformulating the problem as a system of integral equations, reducing error accumulation over long-time simulations and improving computational efficiency. Key contributions include exponential convergence rates for smooth solutions, robustness under oscillatory conditions, and an inherently parallelizable structure, enabling scalable computation for large-scale problems. Additionally, the method introduces a barycentric formulation of the shifted Gegenbauer–Gauss (SGG) quadrature to ensure high accuracy and stability for relatively low Péclet numbers. This approach simplifies calculations of integrals, making the method faster and more reliable for diverse problems. Numerical experiments presented validate the method’s superior performance over traditional techniques, such as finite difference, finite element, and spline-based methods, achieving near-machine precision with significantly fewer mesh points. These results demonstrate its potential for extending to higher-dimensional problems and diverse applications in computational mathematics and engineering. The method’s fusion of spectral precision and integral reformulation marks a significant advancement in numerical PDE solvers, offering a scalable, high-fidelity alternative to conventional time-stepping techniques. Full article

In this work, we present the procedure to obtain exact spherical shape functions for finite element modeling applications, without resorting to any kind of approximation, for generic prismatic spherical elements and for the case of spherical six-node tri-rectangular and eight-node quadrangular spherical prisms. The proposed spherical shape functions, given in explicit analytical form, are expressed in geographic coordinates, namely colatitude, longitude and distance from the center of the sphere. We demonstrate that our analytical shape functions satisfy all the properties required by this class of functions, deriving at the same time the analytical expression of the Jacobian, which allows us changes in coordinate systems. Within the perspective of volume integration on Earth, entering a variety of geophysical and geodetic problems, as for mass change contribution to gravity, we consider our analytical expression of the shape functions and Jacobian for the six-node tri-rectangular and eight-node quadrangular right spherical prisms as reference volumes to evaluate the volume of generic spherical triangular and quadrangular prisms over the sphere; volume integration is carried out via Gauss–Legendre quadrature points. We show that for spherical quadrangular prisms, the percentage volume difference between the exact and the numerically evaluated volumes is independent from both the geographical position and the depth and ranges from 10⁻³ to lower than 10⁻⁴ for angular dimensions ranging from 1° × 1° to 0.25° × 0.25°. A satisfactory accuracy is attained for eight Gauss–Legendre quadrature points. We also solve the Poisson equation and compare the numerical solution with the analytical solution, obtained in the case of steady-state heat conduction with internal heat production. We show that, even with a relatively coarse grid, our elements are capable of providing a satisfactory fit between numerical and analytical solutions, with a maximum difference in the order of 0.2% of the exact value. Full article

This work presents the development of a functional real-time SLAM system designed to enhance the perception capabilities of an Automated Guided Vehicle (AGV) using only a 2D LiDAR sensor. The proposal aims to address recurring gaps in the literature, such as the need for low-complexity solutions that are independent of auxiliary sensors and capable of operating on embedded platforms with limited computational resources. The system integrates scan alignment techniques based on the Iterative Closest Point (ICP) algorithm. Experimental validation in a controlled environment indicated better performance using Gauss–Newton optimization and the point-to-plane metric, achieving pose estimation accuracy of 99.42%, 99.6%, and 99.99% in the position ($x$, $y$) and orientation ($\theta$) components, respectively. Subsequently, the system was adapted for operation with data from the onboard sensor, integrating a lightweight graphical interface for real-time visualization of scans, estimated pose, and the evolving map. Despite the moderate update rate, the system proved effective for robotic applications, enabling coherent localization and progressive environment mapping. The modular architecture developed allows for future extensions such as trajectory planning and control. The proposed solution provides a robust and adaptable foundation for mobile platforms, with potential applications in industrial automation, academic research, and education in mobile robotics. Full article

This article proposes a method to solve nonlinear optimal control problems with arbitrary performance indices and terminal constraints, which is based on the neighboring extremal method and Gauss pseudospectral collocation. Firstly, a quadratic performance index is formulated, which minimizes the second-order variation of the nonlinear performance index and fully considers the deviations in initial states and terminal constraints. Secondly, the first-order necessary conditions are applied to derive the perturbation differential equations involving deviations in state and costate variables. Therefore, a quadratic optimal control problem is formulated, which is subject to such perturbation differential equations. Thirdly, the Gauss pseudospectral collocation is used to transform the differential and integral operators into algebraic operations. Therefore, an analytical solution of the control correction can be successfully derived in the polynomial space, which comes close to the optimal solution. This method has a fast computation speed and low computational complexity due to the discretization at orthogonal points, making it suitable for online applications. Finally, some simulations and comparisons with the optimal solution and other typical methods have been carried out to evaluate the performance of the method. Results show that it not only performs well in computational efficiency and accuracy but also has great adaptability and optimality. Moreover, Monte Carlo simulations have been conducted. The results demonstrate that it has strong robustness and excellent performance even in highly dispersed environments. Full article

To ensure the sustainable development of the surrounding environment and the sustainable operation of landfills, detecting landfill leakage is of great significance. In landfills lacking a leakage monitoring system, the inability to detect and locate damaged High-Density Polyethylene (HDPE) membranes can lead to the contamination of soil and groundwater by landfill leachate. To address this issue, this study proposes a resistivity tomography inversion model based on the external-electrode power supply mode. Utilizing the resistivity difference between the leakage zone and the surrounding soil, electrodes are arranged symmetrically for both power supply and measurement. Upon applying direct current (DC) excitation, potential data are collected, with the finite volume method employed for inversion and the Gauss–Newton method integrated with an adaptive particle swarm optimization algorithm for parameter fitting. Experimental results show that the combined algorithm provides better clarity in edge recognition of low-resistance models compared with single algorithms. The maximum deviation between inferred leakage coordinates and the actual location is 10.1 cm, while the minimum deviation is 6.4 cm, satisfying engineering requirements. This method can effectively locate point sources and line sources, providing an accurate solution for subsequent leakage point filling and improving repair efficiency. Full article

A pseudo-spectral control algorithm based on adaptive Gauss collocation point reconstruction is proposed to efficiently solve the optimal dynamics control problem of industrial robots. A mathematical model for the kinematic relationship and dynamic optimization control of industrial robots has been established. On the basis of deriving the Legendre–Gauss collocation formula, a two-stage adaptive Gauss collocation strategy for industrial robot dynamics control variables was designed to improve the dynamics optimization control effect of industrial robot by improving the solution efficiency of constrained optimization problems. The results show that compared with the control variable parameterization method and the traditional Gaussian pseudo-spectral method, the proposed dynamic optimal control method based on an adaptive Gaussian point reconstruction algorithm can effectively improve the solving time and efficiency of constrained optimization problems, thereby further enhancing the dynamic optimization control and trajectory tracking effect of industrial robots. Full article

Our paper introduces a new theoretical framework called the Fractional Einstein–Gauss–Bonnet scalar field cosmology, which has important physical implications. Using fractional calculus to modify the gravitational action integral, we derived a modified Friedmann equation and a modified Klein–Gordon equation. Our research reveals non-trivial solutions associated with exponential potential, exponential couplings to the Gauss–Bonnet term, and a logarithmic scalar field, which are dependent on two cosmological parameters, m and ${\alpha }_0=t_0{H}_0$ and the fractional derivative order $\mu$. By employing linear stability theory, we reveal the phase space structure and analyze the dynamic effects of the Gauss–Bonnet couplings. The scaling behavior at some equilibrium points reveals that the geometric corrections in the coupling to the Gauss–Bonnet scalar can mimic the behavior of the dark sector in modified gravity. Using data from cosmic chronometers, type Ia supernovae, supermassive Black Hole Shadows, and strong gravitational lensing, we estimated the values of m and ${\alpha }_0$, indicating that the solution is consistent with an accelerated expansion at late times with the values ${\alpha }_0=1.38\pm 0.05$, $m=1.44\pm 0.05$, and $\mu =1.48\pm 0.17$ (consistent with ${\mathrm{\Omega }}_{m,0}=0.311\pm 0.016$ and $h=0.712\pm 0.007$), resulting in an age of the Universe $t_0=19.0\pm 0.7$ [Gyr] at 1$\sigma$ CL. Ultimately, we obtained late-time accelerating power-law solutions supported by the most recent cosmological data, and we proposed an alternative explanation for the origin of cosmic acceleration other than $\mathrm{\Lambda }$CDM. Our results generalize and significantly improve previous achievements in the literature, highlighting the practical implications of fractional calculus in cosmology. Full article

Joint localization and synchronization (JLAS) is a technology that simultaneously determines the spatial locations of user nodes and synchronizes the clocks between user nodes (UNs) and anchor nodes (ANs). This technology is crucial for various applications in wireless sensor networks. Existing solutions for JLAS are either computationally demanding or not resilient to noise. This paper addresses the challenge of localizing and synchronizing a mobile user node in broadcast-based JLAS systems using sequential one-way time-of-arrival (TOA) measurements. The AN position uncertainty is considered along with clock offset and skew. Two redundant variables that couple the unknowns are introduced to pseudo-linearize the measurement equation. In projecting the equation to the nullspace spanned by the coefficients of the redundant variables, the affection of them can be eliminated. While the closed-form projection solution provides an initial point for iteration, it is suboptimal and may not achieve the Cramér-Rao lower bound (CRLB) when noise or AN position error is relatively large. To improve performance, we propose a novel robust iterative solution (RIS) formulated through factor graphs and developed via message passing. The RIS outperforms the common Gauss–Newton iteration, especially in high-noise scenarios. It exhibits a lower root mean-square error (RMSE) and a higher probability of converging to the optimal solution, while maintaining manageable computational complexity. Both analytical results and numerical simulations validate the superiority of the proposed solution in terms of performance, resilience, and computational load. Full article

We apply known special functions from the literature (and these include the Fox $\mathbb{H}–$function, the exponential function, the Mittag-Leffler function, the Gauss Hypergeometric function, the Wright function, the $\mathbb{G}–$function, the Fox–Wright function and the Meijer $\mathbb{G}–$function) and fuzzy sets and distributions to construct a new class of control functions to consider a novel notion of stability to a fractional-order system and the qualified approximation of its solution. This new concept of stability facilitates the obtention of diverse approximations based on the various special functions that are initially chosen and also allows us to investigate maximal stability, so, as a result, enables us to obtain an optimal solution. In particular, in this paper, we use different tools and methods like the Gronwall inequality, the Laplace transform, the approximations of the Mittag-Leffler functions, delayed trigonometric matrices, the alternative fixed point method, and the variation of constants method to establish our results and theory. Full article

This paper proposes an original approach to calculating integrals of rapidly oscillating functions, based on Levin’s algorithm, which reduces the search for an anti-derivative function to solve an ODE with a complex coefficient. The direct solution of the differential equation is based on the method of integrating factors. The reduction in the original integration problem to a two-stage method for solving ODEs made it possible to overcome the instability that arises in the standard (in the form of solving a system of linear algebraic equations) approach to the solution. And due to the active use of Chebyshev interpolation when using the collocation method on Gauss–Lobatto grids, it is possible to achieve high speed and stability when taking into account a large number of collocation points. The presented spectral method of integrating factors is both flexible and reliable and allows for avoiding the ambiguities that arise when applying the classical method of collocation for the ODE solution (Levin) in the physical space. The new method can serve as a basis for solving ordinary differential equations of the first and second orders when creating high-efficiency software, which is demonstrated by solving several model problems. Full article

Urban autonomous vehicles on city roads are subject to various constraints when changing lanes, and commonly used trajectory planning methods struggle to describe these conditions accurately and directly. Therefore, generating accurate and adaptable trajectories is crucial for safer and more efficient trajectory planning. This study proposes an optimal control model for local path planning that integrates dynamic vehicle constraints and boundary conditions into the optimization problem’s constraint set. Using the lane-changing scenario as a basis, this study establishes environmental and collision avoidance constraints during driving and develops a performance metric that optimizes both time and turning angle. The study employs the Gauss pseudo-spectral method to continuously discretize the state and control variables, converting the optimal control problem into a nonlinear programming problem. Using numerical solutions, variable control and state trajectories that satisfy multiple constraint conditions while optimizing the performance metric are generated. The study employs two weights in the experiment to evaluate the method’s performance, and the findings demonstrate that the proposed method guarantees safe obstacle avoidance, is stable, and is computationally efficient at various interpolation points compared to the Legendre pseudo-spectral method (LPM) and the Shooting method. Full article

One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss–Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme’s accuracy and compare it to existing findings. Full article