Gauss-Legendre Quadrature Rule Research Articles

A new variational integrator for constrained mechanical system dynamics

A new variational integrator is proposed to solve constrained mechanical systems. The main distinguishing feature of the present integrator comes from the distinct discretization of Lagrangians based on the Hamilton's principle in its most general form. Specifically, Hermite interpolation is used for discrete positions, which provides at least C1 continuity for generalized coordinates. The velocities and momentums are interpolated using quadratic polynomials for the consistency, such that the kinematic relation between velocities and positions can be exactly satisfied. Meanwhile, the Gauss-Legendre quadrature rule is employed to guarantee the accuracy of discrete Lagrange equations. To tackle constrained mechanical systems, a coordinate partition approach is used to eliminate the constraint equations. The local incremental rotation vector is exploited to get rid of rotation singularities in spatial problems. Moreover, an adaptive stepsize strategy is implemented to improve the efficiency. The strengths of the new integrator lie in the accessible large step sizes in the simulation and its global second-order accuracy for positions as well as velocities. Several examples are performed and analyzed to validate its accuracy and capabilities.

Theoretical error estimates for computing the matrix logarithm by Padé-type approximants

In this article, we focus on the error that is committed when computing the matrix logarithm using the Gauss–Legendre quadrature rules. These formulas can be interpreted as Padé approximants of a suitable Gauss hypergeometric function. Empirical observation tells us that the convergence of these quadratures becomes slow when the matrix is not close to the identity matrix, thus suggesting the usage of an inverse scaling and squaring approach for obtaining a matrix with this property. The novelty of this work is the introduction of error estimates that can be used to select a priori both the number of Legendre points needed to obtain a given accuracy and the number of inverse scaling and squaring to be performed. We include some numerical experiments to show the reliability of the estimates introduced.

A numerical method for Ψ-fractional integro-differential equations by Bell polynomials

In this work, we focus on a class of Ψ− fractional integro-differential equations (Ψ-FIDEs) involving Ψ-Caputo derivative. The objective of this paper is to derive the numerical solution of Ψ-FIDEs in the truncated Bell series. Firstly, Ψ-FIDEs by using the definition of Ψ− Caputo derivative is converted into a singular integral equation. Then, a computational procedure based on the Bell polynomials, Gauss-Legendre quadrature rule, and collocation method is developed to effectively solve the singular integral equation. The convergence of the approximation obtained in the presented strategy is investigated. Finally, the effectiveness and superiority of our method are revealed by numerical samples. The results of the suggested approach are compared with the results obtained by extended Chebyshev cardinal wavelets method (EChCWM).

Angular distribution measurements of neutron elastic scattering on natural carbon

The angular distributions of neutron elastic scattering on natural carbon were studied in the fast neutron energy region between 1 and 8 MeV. The experiments were carried out at the white neutron source of the Geel Electron Linear Accelerator (GELINA) facility by using thin and thick natural carbon samples. This work demonstrates the need for using thin samples to avoid strong multiple scattering effects. Neutrons and γ rays from scattering were detected using the ELastic and Inelastic Scattering Array (ELISA), a setup consisting of 32 liquid organic scintillators. The n−γ separation was achieved via pulse-shape analysis. For each sample a different approach in methodology is studied, one based on a global response function model and another one based on a per-detector model. The detectors are placed at eight different detection angles between 16∘ and 164∘ with respect to the neutron beam direction, allowing the simultaneous calculation of both the differential and the integral cross section by implementing the Gauss-Legendre quadrature rule. The neutron flux was measured with a U235 ionization chamber. The angular distributions were extracted relative to the U235(n,f) cross section. The results are compared with other experimental data available in the EXFOR library, along with the most recent nuclear data evaluations. The angle-integrated cross sections are in excellent agreement with the nuclear data evaluations and for the angular distributions, ENDF/B-VIII.0 is better reproducing the experimental data in all eight detection angles. Published by the American Physical Society 2024

A robust optimisation approach for the 2D-VO fractional optimal control problems

This paper provides a new numerical scheme with the help of discrete Krawtchouk polynomials (DKPs), the Ritz method, and the two-dimensional (2D) Gauss-Legendre quadrature rule for dealing with two-dimensional variable-order (VO) fractional optimal control problems. For this aim, we construct the derivative operational matrices of DKPs. Also, we impose the conditions into the approximation of state and control functions. Then, we expand the functions in the problem given DKPs and operational matrices. Meanwhile, DKPs and the 2D Gauss-Legendre quadrature rule evaluate the performance index function. According to the numerical algorithm process, the proposed problem is reduced to a system of algebraic equations. The chosen polynomial as basis functions and the Ritz method provide a powerful and flexible scheme. Also, we discuss the error of the VO-fractional derivative of the approximate solution and performance index. At last, we include several illustrative examples to indicate the validity and applicability of the present technique.

Fast Zernike fitting of freeform surfaces using the Gauss-Legendre quadrature

Zernike polynomial orthogonality, an established mathematical principle, is leveraged with the Gauss-Legendre quadrature rule in a rapid novel approach to fitting data over a circular domain. This approach provides significantly faster fitting speeds, in the order of thousands of times, while maintaining comparable error rates achieved with conventional least-square fitting techniques. We demonstrate the technique for fitting mid-spatial-frequencies (MSF) prevalent in small-tool-manufacturing typical of aspheric and freeform optics that are poised to soon permeate a wide range of optical technologies.

Optimal solution of nonlinear 2D variable-order fractional optimal control problems using generalized Bessel polynomials

This study aims to propose a new optimization method based on the generalized Bessel polynomials (GBPs) as a class of basis functions for a category of nonlinear two-dimensional variable-order fractional optimal control problems (N-2D-VOFOCPs) involved in fractional-order dynamical systems and Caputo derivatives. For the optimal solution of such problems, the optimization method is developed on the basis of operational matrices (OMs) scheme of derivatives, 2D Gauss–Legendre quadrature rule, and Lagrange multiplier technique. The state and control functions are expanded in terms of the GBPs to reduce the complexity of these problems. The proposed method focuses on a system of nonlinear algebraic equations in the process of finding solution to the problems. The convergence of the method based on GBPs is proved, and the accuracy of the method is analyzed by solving several examples.

Cross section measurements of neutron elastic and inelastic scattering on Fe54

Scattering experiments were performed at the Geel Electron Linear Accelerator (GELINA) to determine the differential cross sections of neutron elastic and inelastic scattering on Fe54, using a highly enriched sample. Neutron energies were calculated via the time-of-flight technique. The scattered neutrons were detected using the ELastic and Inelastic Scattering Array (ELISA), a spectrometer consisting of 32 liquid organic scintillators, able to separate neutron and photon induced events via pulse-shape analysis. The detectors are mounted at eight specific angles with respect to the neutron beam direction to allow the concurrent calculation of both the differential and the integral cross section using the Gauss-Legendre quadrature rule. For elastic scattering, angular distributions and integral cross sections were produced in the energy range from 1 to 8 MeV, whereas for inelastic scattering, partial differential and angle integrated cross sections were estimated from the first excited state of Fe54 in the energy range from 2.5 to 5.5 MeV. The results are compared with the available experimental and evaluated nuclear data, as well as with theoretical calculations performed with the TALYS 1.9 and EMPIRE 3.2 codes. Published by the American Physical Society 2024

An application of Hayashi's inequality in numerical integration

Abstract This study systematically develops error estimates tailored to a specific set of general quadrature rules that exclusively incorporate first derivatives. Moreover, it introduces refined versions of select generalized Ostrowski’s type inequalities, enhancing their applicability. Through the skillful application of Hayashi’s celebrated inequality to specific functions, the provided proofs underpin these advancements. Notably, this approach extends its utility to approximate integrals of real functions with bounded first derivatives. Remarkably, it employs Newton-Cotes and Gauss-Legendre quadrature rules, bypassing the need for stringent requirements on higher-order derivatives.

Numerical Solution of Fractional Order Integro-Differential Equations via Müntz Orthogonal Functions

In this paper, we derive a spectral collocation method for solving fractional-order integro-differential equations by using a kind of Müntz orthogonal functions that are defined on 0,1 and have simple and real roots in this interval. To this end, we first construct the operator of Riemann–Liouville fractional integral corresponding to this kind of Müntz functions. Then, using the Gauss–Legendre quadrature rule and by employing the roots of Müntz functions as the collocation points, we arrive at a system of algebraic equations. By solving this system, an approximate solution for the fractional-order integro-differential equation is obtained. We also construct an upper bound for the truncation error of Müntz orthogonal functions, and we analyze the error of the proposed collocation method. Numerical examples are included to demonstrate the validity and accuracy of the method.

The numerical solution of a time-delay model of population growth with immigration using Legendre wavelets

The paper addresses a computational method to simulate more accurate models for population growth with immigration, focusing on integral equations (IEs) featuring a delay parameter in the time variable. The proposed method utilizes Legendre wavelets within the Galerkin scheme as a orthonormal basis. Legendre wavelets are known for their localized functions, offering suitable precision and stability in simulating time-delay biological models. This approach employs the composite Gauss-Legendre (CGL) quadrature rule to compute integrals appeared in the scheme. An error bound analysis demonstrates a convergence rate of order 2−Mk. Additionally, various numerical examples are presented to show the efficiency, accuracy and validate the theoretical error estimate of the novel technique.

Statistical Design for Monitoring Process Mean on Modified EWMA Control Chart based on Autocorrelated Data

This research endeavor is focused on establishing explicit formulas for the computation of the average run length (ARL) within the context of a moving average process characterized by exogenous variables, denoted as MAX(q,r), and subjected to exponential white noise. Additionally, we aim to conduct a comparative analysis of their performance against the exponentially weighted moving average (EWMA) and the modified exponentially weighted moving average (modified EWMA) methodologies. The evaluation of their performance will be based on metrics such as the absolute percentage relative error (APRE) and the relative mean index (RMI). Furthermore, we undertake a rigorous assessment of the accuracy of these explicit formulas in relation to ARL by considering CPU time, utilizing the numerical integral equation (NIE) method derived through the application of the Gauss-Legendre quadrature rule. This comparative evaluation is carried out for both control chart methodologies. To ascertain the efficacy of our explicit formulas approach, we apply it to two distinct datasets. The first dataset pertains to the closing price of natural gas, with the crude oil WTI price serving as the exogenous variable. The second dataset encompasses the closing stock price of KTB Public Company Limited, with daily foreign exchange rates for USD/JPY and EUR/USD as the exogenous variables. The results of applying the ARL based on the explicit formulas to these two datasets demonstrate that, under these conditions, the modified EWMA control chart outperforms the EWMA control chart.

Deep reduced-order least-square method—A parallel neural network structure for solving beam problems

In this paper, a novel physics-informed neural networks (PINNs) called deep reduced-order least-square (ROLS) is proposed. The deep ROLS is a combination of reduced-order and least-square methods in the context of PINN methodology. The key idea of the deep ROLS is to convert a higher-order PDE to a system of lower-order PDEs by using primary and secondary variables, then the loss function constituted by integrals of corresponding squared residuals over the problem domain is minimized. Specifically, the squared residuals are established based on L2 norm of respective reduced-order PDE terms, and the Gauss–Legendre quadrature rule is applied to approximate their integrals. A multi-network structure is also designed, in which each sub-network serves as one field variable corresponding to each reduced-order PDE. Moreover, this work proposes a scheme to handle both essential and natural boundary conditions (BCs) directly. The deep ROLS demonstrates its better advantages in comparison to original PINN method in terms of solution accuracy, computational cost and data efficiency for several beam bending problems having continuous and discontinuous solutions.

The direct meshless local Petrov–Galerkin technique with its error estimate for distributed-order time fractional Cable equation

Distributed-order fractional calculus is a quickly growing concept of the more general area of fractional calculus that has significant and extensive usage for designing complex systems. This work is used the direct meshless local Petrov–Galerkin (DMLPG) technique for the numerical solution of the distributed-order time fractional Cable equation. DMLPG implements a generalized moving least square (GMLS) process to discretize the equation in space variables. By using this scheme, the test function is approximated via the values at nodes, directly. Thus, this algorithm passes integration with the MLS shape functions substituting with a more inexpensive integration than polynomials. Here, the distributed integral is discretized by the M-point Gauss–Legendre quadrature rule. Then, the finite difference scheme is applied to approximate the fractional derivative discretization. Also, the unconditionally stability and rate of convergence O(τ2−max{α,β}) of the time-discrete technique are demonstrated. Moreover, the current method converts the problem into a system of linear algebraic equations. To demonstrate the capability and flexibility of our scheme, some examples with different geometric domains are supposed in two-dimensional cases.

Probabilistic response analysis of nonlinear vibro-impact systems with two correlated Gaussian white noises

The response distributions of a dynamic system under the effect of correlation noise not only demonstrated a Gaussian symmetric distribution, but also exhibited a non-Gaussian skewed distribution. Thus, correlation noise can induce more complex dynamic phenomena in the response of a nonlinear system. The purpose of this study is to explore the stochastic response and bifurcation of nonlinear vibro-impact systems driven by correlated additive and multiplicative Gaussian white noises using the path integral (PI) method. First, we derived an equivalent correlated stochastic system without collision conditions in the new ancillary phase space using the non-smooth Ivanov transform method. We then used Novikov’s theorem to derive an effective Fokker–Planck–Kolmogorov equation for the equivalent correlated stochastic system. Second, we used PI technology, which combines the three-point Gauss–Legendre quadrature rule to solve the effective Fokker–Planck–Kolmogorov equation and obtained the probability density functions (PDFs) of the equivalent correlated stochastic system. Subsequently, we obtained the PDFs of the original vibro-impact stochastic system using the relationship between the initial state space and the new auxiliary phase space. Finally, the proposed method was used to investigate the transient and stationary responses of a Duffing-Van der Pol Vibro-impact system perturbed by additive and multiplicative random noises. In addition, this technique can also assess the effects of restitution coefficients and barrier position on stochastic P-bifurcation. The accuracy of this method was verified by comparing it with a Monte Carlo simulation.

Quantum algorithm for matrix logarithm by integral formula

In scientific computing, one can find a wide application of the matrix-vector product f(A)b. Recently, a quantum algorithm that computes the state \(|f\rangle \) corresponding to f(A)b has been proposed in Takahira et al. (Quantum Inf Comput 20(1/2):14–36, 2020). However, this important algorithm can not be directly applied to the matrix logarithm, which is one of the significant matrix functions. In this paper, we propose an original quantum algorithm to compute the state \(|f\rangle = \log (A)|b\rangle / \Vert \log (A)|b\rangle \Vert \), via the integral representation of \(\log (A)\) and the Gauss–Legendre quadrature rule, using LCU method and block-encoding technique as subroutines.

Exact Evaluation of Time-Domain Physical Optics Integral for High-Order Triangles

In this work, a method for the exact evaluation of time domain physical optics (PO) integral on high-order triangles is presented. The presented method is based on the recently developed method for quadratic triangles. Specifically, the PO integral on the surface of the high-order triangle is reduced to a line integral on the intersection of the high-order triangle and the plane formed by the incidence and observation directions in barycentric coordinates of the triangle. Then, the intersecting curve is parametrized in terms of one of the barycentric coordinates of the high-order triangle. At last, the line integral is evaluated exactly using the suitable order Gauss Legendre quadrature rule (GLQR). The validity of the proposed method is demonstrated via analyzing transient scattering from a unit sphere and a flower-shaped scatterer modeled with high-order triangles using PO approximation.

Discretization of Fractional Order Operator in Delta Domain

The fractional order operator is the backbone of the fractional order system (FOS). The fractional order operator (FOO) is generally represented as s^(±μ) (0<μ<1). Discrete time FOS can be obtained through the discretization of the fractional order operator. The FOO is the general form of either fractional order differentiator (FOD) or integrator (FOI) depending upon the values of μ. Out of the two discretization methods, direct discretization outperforms the method of indirect discretization. The mapping between the continuous time and discrete time domain is done with the development of generating function. Continuous fraction expansion (CFE) is used expand the generating function for the rational approximation of the FOO. There is an inherent problem associated with the discretization of FOO in discrete z-domain particularly at very fast sampling rate. In the other hand, discretization using delta operator parameterization provides the continuous time and discrete time results in hand to hand, when the continuous time systems are sampled at very fast sampling rate and circumventing the problem with shift operator parameterization at fast sampling rate. In this work, a new generating function is proposed to discretize the FOO using the Gauss-Legendre 3-point quadrature rule and generating function is expanded using the CFE to form rational approximation of the FOO in delta domain. The benchmark fractional order systems are considered in this work for the simulation purpose and comparison of results are made to prove the efficacy of the proposed method using MATLAB.

A discrete collocation method based on the radial basis functions for solving system of integral equations of the second kind

In this paper, the system of integral equations of the second kind is investigated by using a discrete collocation method based on the redial basis functions (RBFs). To construct the discrete RBFs-collocation method, the Gauss–Legendre quadrature rule is adopted for the numerical integration. The error and convergence of the algorithm are given strictly. The efficiency of the approach will be shown by applying the procedure on some ill-posed and well-posed prototype examples.

A numerical method based on the piecewise Jacobi functions for distributed-order fractional Schrödinger equation

In this work, the distributed-order time fractional version of the Schrödinger problem is defined by replacing the first order derivative in the classical problem with this kind of fractional derivative. The Caputo fractional derivative is employed in defining the used distributed fractional derivative. The orthonormal piecewise Jacobi functions as a novel family of basis functions are defined. A new formulation for the Caputo fractional derivative of these functions is derived. A numerical method based upon these piecewise functions together with the classical Jacobi polynomials and the Gauss–Legendre quadrature rule is constructed to solve the introduced problem. This method converts the mentioned problem into an algebraic problem that can easily be solved. The accuracy of the method is examined numerically by solving some examples.