Search Results (55)

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

A multiplexing method for orbital angular momentum (OAM) beams was proposed. The aperture size as a new information carrier was provided, and it could be modulated by the external variable aperture. The field of the beams propagating through turbulence was derived and discretized with Gauss–Legendre quadrature formulas. Based on this, the demultiplexing method was improved, and the beam OAM states, amplitude, Gaussian spot radius and aperture radius were decoded. Moreover, the influence of turbulence on the multiplexing parameters was also analyzed, and the decoding precision of the aperture radius was higher than that of other parameters. The aperture radius was recommended as an extra carrier for multiplexing communication. This study provides a simple method to modulate the information carried by OAM beams, and it has promising applications in large capacity laser communication. Full article

This study investigates the propagation of shear horizontal transverse waves in a functionally graded piezoelectric half-space (FGPHS), where the material properties vary linearly and quadratically. The analysis focuses on deriving and understanding the dispersion characteristics of such waves in in-homogeneous media. The WKB approximation method is employed to obtain the dispersion relation analytically, considering the smooth variation of material properties. To validate and study the wave behavior numerically, two advanced techniques were utilized: the Semi-Analytical Finite Element with Perfectly Matched Layer (SAFE-PML) and the Semi-Analytical Infinite Element (SAIFE) method incorporating a (1/r) decay model to simulate infinite media. The numerical implementation uses the Rayleigh–Ritz method to discretize the wave equation, and Gauss 3-point quadrature is applied for efficient numerical integration. The dispersion curves are plotted to illustrate the wave behavior in the graded piezoelectric medium. The results from SAFE-PML and SAIFE are in excellent agreement, indicating that these techniques effectively model the shear horizontal transverse wave propagation in such structures. This study also demonstrates that combining finite and infinite element approaches provides accurate and reliable simulation of wave phenomena in functionally graded piezoelectric materials, which has applications in sensors, actuators, and non-destructive testing. Full article

Lewis’ theory of multiple scattering has been modeled as a random walk on a unit sphere for calculating the multiple scattering angular distribution of charged particles, which is more intuitive and mathematically simpler. This formalism can lead to the Goudsmit–Saunderson theory and the Lewis theory of multiple scattering angular distribution, thus providing an easier-to-understand framework to unify both the Goudsmit–Saunderson and the Lewis theories. This new random walk method eliminates the need for integro-differential expansions in Lewis theory and is faster at calculating multiple scattering angular distributions, reducing the required Legendre series terms by 80% at small step (path) length (<20) and providing much greater calculation efficiency. Crucially, the random walk formalism explicitly preserves spherical symmetry by treating angular deflections as steps on a unit sphere, enabling the efficient sampling of scattering events while maintaining accuracy. Further, a robust algorithm for numerically calculating multiple scattering angular distributions of electrons based on the Goudsmit–Saunderson and Lewis theories has been developed. Partial wave elastic scattering differential cross-sections, generated with the program ELSEPA, have been used in the calculations. A two-point Gauss–Legendre quadrature method is used to calculate the Legendre coefficients (multiple scattering moments). Full article

In this paper, we propose a neural network method to solve time-fractional diffusion equations with Dirichlet boundary conditions by using a combination of machine learning techniques and Method of Lines. We first used the Method of Lines to discretize the equation in the space domain while keeping the time domain continuous, and represent the solution of the diffusion equation using a neural network. Then we used Gauss–Jacobi quadrature to approximate the fractional derivative in the time domain, thereby obtaining the loss function for the neural network. We used TensorFlow to carry out the gradient descent process to train this neural network. We conducted numerical tests in 1D and 2D cases and compared the results with the exact solutions. The numerical tests showed that this method is effective and easy to manipulate for many time-fractional diffusion problems. Full article

This paper describes a Krylov subspace iterative method designed for solving linear systems of equations with a large, symmetric, nonsingular, and indefinite matrix. This method is tailored to enable the evaluation of error estimates for the computed iterates. The availability of error estimates makes it possible to terminate the iterative process when the estimated error is smaller than a user-specified tolerance. The error estimates are calculated by leveraging the relationship between the iterates and Gauss-type quadrature rules. Computed examples illustrate the performance of the iterative method and the error estimates. Full article

Conventional adaptive beamformers usually suffer from serious performance degradation when the receive array is imperfect and unknown sporadic interferences appear. To enhance robustness against array imperfections and simultaneously suppress sporadic interferences, this paper studies robust adaptive beamforming (RAB) with accurate sidelobe level (SLL) control, where the imperfect array steering vector (SV) is expressed as a spherical uncertainty set. Under the maximum signal-to-interference-plus-noise ratio (SINR) criterion and robust SLL constraints, we formulate the resultant RAB into a second-order cone programming problem, which is computationally prohibitive due to numerous robust quadratic SLL constraints. To tackle this issue, we provide a subspace approximation-based method to approximate the whole sidelobe space, thus replacing all robust SLL constraints with a single subspace constraint. Moreover, we leverage the Gauss–Legendre quadrature-based scheme to generate the sidelobe space in a computationally efficient manner. Additionally, we give an explicit approach for determining the norm upper bound of SV uncertainty sets under various imperfection scenarios, addressing the challenge of obtaining this upper bound in practice.Simulation results showed that the proposed subspace approximation-based RAB beamformer had a better SINR performance than typical counterparts and was much more computationally efficient. Full article

A damped iterative explicit guidance (DIEG) algorithm is proposed to address the problem of the insufficient convergence of classical explicit guidance methods in the event of thrust drop faults in multistage rockets. Based on the iterative guidance mode (IGM) and powered explicit guidance (PEG), this method is enhanced in three aspects: (1) an accurate transversality condition is derived and applied in the dimension-reduction framework instead of using a simplified assumption; (2) the Gauss–Legendre quadrature formula (GLQF) is adopted to increase the accuracy of the method by addressing the issue of excessive errors in calculating thrust integration using linearization methods based on a small quantity assumption under fault conditions; and (3) a damping factor for solving the time-to-go is introduced to avoid the chattering phenomenon and enhance convergence. A numerical simulation was conducted in single- and multistage mission scenarios by gradually reducing the engine thrust to compare the performance of DIEG and PEG. The results show that DIEG has a much larger convergence range than PEG and has fuel optimality similar to that of the optimization method in most fault scenarios. Finally, the robustness of DIEG under various deviations is verified through Monte Carlo simulation. Full article

Forward modeling the magnetic effects of an inferred source is the basis of magnetic anomaly inversion for estimating subsurface magnetization parameters. This study uses numerical least-squares Gauss–Legendre quadrature (GLQ) integration to evaluate the magnetic potential, anomaly, and gradient components of a cylindrical prism element. Relative to previous studies, it quantifies for the first time the magnetic gradient components, enabling their applications in the interpretation of cylindrical bodies. A comparison of this method to other methods of evaluating the vertical component of the magnetic field associated with a full cylinder shows that it has comparable to improved performance in computational accuracy and speed. Based on the developed theory, a conceptual design is presented for an instrument to measure the magnetic gradient effects of subsurface material in the vicinity of a borehole. The significance of this instrument relative to conventional borehole magnetometers is in its ability to determine the azimuthal directions of magnetic sources within the borehole environment. Full article

This study introduces the Lindley Stochastic Frontier Analysis—LSFA model, a novel approach that incorporates the Lindley distribution to enhance the flexibility and accuracy of efficiency estimation. The LSFA model is compared against traditional SFA models, including the half-normal, exponential, and gamma models, using advanced numerical methods such as the Gauss–Hermite Quadrature, Monte Carlo Integration, and Simulated Maximum Likelihood Estimation for parameter estimation. Simulation studies revealed that the LSFA model outperforms in scenarios involving small sample sizes and complex, skewed distributions, particularly those characterized by gamma distributions. In contrast, traditional models such as the half-normal model perform better in larger samples and simpler settings, while the gamma model is particularly effective under exponential inefficiency distributions. Among the numerical techniques, the Gauss–Hermite Quadrature demonstrates a strong performance for half-normal distributions, the Monte Carlo Integration offers consistent results across models, and the Simulated Maximum Likelihood Estimation shows robustness in handling gamma and Lindley distributions despite higher errors in simpler cases. The application to a banking dataset assessed the performance of 12 commercial banks pre-COVID-19 and during COVID-19, demonstrating LSFA’s superior ability to handle skewed and intricate data structures. LSFA achieved the best overall reliability in terms of the root mean square error and bias, while the gamma model emerged as the most accurate for minimizing absolute and percentage errors. These results highlight LSFA’s potential for evaluating efficiency during economic shocks, such as the COVID-19 pandemic, where data patterns may deviate from standard assumptions. This study highlights the advantages of the Lindley distribution in capturing non-standard inefficiency patterns, offering a valuable alternative to simpler distributions like the exponential and half-normal models. However, the LSFA model’s increased computational complexity highlights the need for advanced numerical techniques. Future research may explore the integration of generalized Lindley distributions to enhance model adaptability with enriched numerical optimization to establish its effectiveness across diverse datasets. Full article

In this paper, we have established a numerical method for a class of time-fractional and Riesz space distributed-order advection–diffusion equation with time-delay. Firstly, we transform the Riesz space distributed-order derivative term of the diffusion equation into multi-term fractional derivatives by using the Gauss quadrature formula. Secondly, we discretize time by using second-order finite differences, discretize space by using second kind Chebyshev polynomials, and convert the multi-term fractional equation to a system of algebraic equations. Finally, we solve the algebraic equations by the iterative method, and prove the stability and convergence. Moreover, relevant examples are shown to verify the validity of our algorithm. Full article

This paper introduces a novel approach for solving multi-term time-fractional convection–diffusion equations with the fractional derivatives in the Caputo sense. The proposed highly accurate numerical algorithm is based on the barycentric rational interpolation collocation method (BRICM) in conjunction with the Gauss–Legendre quadrature rule. The discrete scheme constructed in this paper can achieve high computational accuracy with very few interval partitioning points. To verify the effectiveness of the present discrete scheme, some numerical examples are presented and are compared with the other existing method. Numerical results demonstrate the effectiveness of the method and the correctness of the theoretical analysis. Full article

In the paper, for hypersingular integral equations with new kernels, a solution is constructed using an approach based on Chebyshev orthogonal polynomials and the principle of contraction mappings. Integrals in hypersingular integral equations are understood in the sense of Hadamard finite-part integrals. The hypersingular integral equations under consideration in some cases of kernels are solved exactly in closed form using the Chebyshev orthogonal polynomial method, and with other kernels by the same method, they are reduced to infinite systems of linear algebraic equations. In addition, hypersingular integral equations with the kernels considered in the article are reduced to finite systems of linear algebraic equations using Gauss–Chebyshev type quadrature formulas. To assess the effectiveness of the two methods, a comparative analysis of the results for hypersingular integral equations with the corresponding kernels is carried out. Full article

A computationally time-efficient method is introduced to implement pressure load to a Finite element model. Hexahedron elements of the Lagrangian family with Gauss–Lobatto nodes and integration quadrature are utilized, where the integration points follow the same sequence as the nodes. This method calculates the equivalent nodal force due to pressure load using a single Hadamard multiplication. The arithmetic operations of this method are determined, which affirms its computational efficiency. Finally, the method is tested with finite element implementation and observed to increase the runtime ratio compared to the conventional method by over 20 times. This method can benefit the implementation of finite element models in fields where computational time is crucial, such as real-time and cyber–physical testbed implementation. Full article

In this paper, we propose a novel approach for solving two-dimensional time-fractional advection–diffusion equations, where the fractional derivative is described in the Caputo sense. The discrete scheme is constructed based on the barycentric rational interpolation collocation method and the Gauss–Legendre quadrature rule. We employ the barycentric rational interpolation collocation method to approximate the unknown function involved in the equation. Through theoretical analysis, we establish the convergence rate of the discrete scheme and show its remarkable accuracy. In addition, we give some numerical examples, to illustrate the proposed method. All the numerical results show the flexible application ability and reliability of the present method. Full article