Gauss-Legendre Quadrature Rule Research Articles

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.

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.

An efficient and accurate chatter prediction method of milling processes with a transition matrix reduction scheme

Efficient and accurate chatter prediction is essential for selection of chatter-free process parameters in order to improve machining productivity and surface quality of the workpiece. This paper proposes a five-point Gaussian quadrature-based chatter prediction method of milling processes together with a transition matrix reduction scheme to improve the computational accuracy and efficiency. The five-point Gauss–Legendre quadrature rule is utilized to approximate the equations of motion of milling systems represented in the form of integral equation. By this means, the convergence rate of the chatter prediction method can be improved. Since the transition matrix describing the relation between the states of current and previous tooth or spindle period is used to evaluate chatter stability, its dimension has a significant effect on the computational efficiency. In this paper, it is theoretically proved that the dimension of the transition matrix constructed by numerical integration-based methods can be reduced by half. Based on this, an efficient scheme is firstly proposed to further decrease the computational time. Typical benchmark examples are employed to verify the proposed method. The results demonstrate that the proposed method with the transition matrix reduction scheme is more efficient and accurate than the existing methods. Meanwhile, experimental validation shows the proposed method can predict the milling chatter accurately.

AN EFFECTIVE AND SIMPLE SCHEME FOR SOLVING NONLINEAR FREDHOLM INTEGRAL EQUATIONS

In this paper, a simple scheme is constructed for finding approximate solution of the nonlinear Fredholm integral equation of the second kind. To this end, the Lagrange interpolation polynomials together with the Gauss-Legendre quadrature rule are used to transform the source problem to a system of nonlinear algebraic equations. Afterwards, the resulting system can be solved by the Newton method. The basic idea is to choose the Lagrange interpolation points to be the same as the points for the Gauss-Legendre integration. This facilitates the evaluation of the integral part of the equation. We prove that the approximate solution converges uniformly to the exact solution. Also, stability of the approximate solution is investigated. The advantages of the method are simplicity, fastness and accuracy which enhance its applicability in practical situations. Finally, we provide some test examples.

Gravity field forward modelling using tesseroids accelerated by Taylor series expansion and symmetry relations

SUMMARY In this study, we developed a new method that can significantly accelerate the forward modelling of gravity fields generated by large-scale tesseroids while keeping the computational accuracy as high as possible. The cost of the high efficiency is that the method only works under the assumptions that (1) all tesseroids in the same latitude band have the same horizontal dimension, (2) the computation points are located at the same surface level and aligned with the horizontal centres of tesseroids and (3) each tesseroid has a constant or linearly varying density. The new method first integrates the kernel function of the Newton’s volume integral analytically in the radial direction to eliminate its dependence on the vertical dimension of the tesseroid, and then expands the integrated kernel function into a Taylor series up to a certain order. Because the Taylor series expansion term of the integrated kernel function is an odd or even function of the difference between the longitudes of the tesseroid and computation point, there exist shifting or swapping symmetry relations among the gravity field of tesseroids. Consequently, the shifting or swapping symmetry is extended to the tesseroids with unequal vertical dimensions. Numerical experiments using the spherical shell model are conducted to verify the effectiveness of the new method. The results show that the computational speed of the new method is about 30 times faster than that of the traditional method, which employs the Gauss–Legendre quadrature rule and a 2-D adaptive subdivision approach, while keeping almost the same computational accuracy. When applying the new method to an ice shell with unequal thicknesses, the results reveal that the relative errors of calculating V, Vz and Vzz are smaller than 10−8, 10−6 and 10−4, respectively if the Taylor series expansion is truncated at order 4, while the computational time consumed by the new method is about 7 times less than that of the traditional method. Finally, the influence of the truncation order on the computational accuracy and the strategies for dividing the latitude band into several parts to further improve the accuracy are discussed.

A regularization of the parametric integral equation system applied to 2D boundary problems for Laplace’s equation with stability evaluation

The paper proposes the regularization of the parametric integral equation system (PIES) for 2D potential problems governed by Laplace’s equation. The regularization eliminates all singularities (weak and strong) in PIES associated with its kernels as a result of their modification and is based on introducing a regularizing function with unknown regularization coefficients. Hence, all integrals appearing in the regularized PIES formulation are non-singular and can be evaluated numerically with a standard low order Gauss–Legendre quadrature rule. Using this formulation, the stability of solutions for different distances between collocation points and quadrature nodes is investigated. The influence of the number of integration and collocation points on the accuracy and stability of solutions for the proposed regularization in comparison with dedicated procedures for evaluation of strongly and weakly singular integrals is also studied. Furthermore, we analyze the impact of the regularization in PIES on the results compared with other methods for evaluating singular integrals in non-regularized PIES.

A new approach for direct discretization of fractional order operator in delta domain

The fractional order system (FOS) comprises fractional order operator. In order to obtain the discretized version of the fractional order system, the first step is to discretize the fractional order operator, commonly expressed as s?m, 0 < m < 1. The fractional order operator can be used as fractional order differentiator or integrator, depending upon the values of . In general, there are two approaches for discretization of fractional order operator, one is indirect method of discretization and another is direct method of discretization. The direct discretization method capitalizes the method of formation of generating function where fractional order operator s?m is expressed as a function of Z in the shift operator parameterization and continued fraction expansion (CFE) method is then utilized to get the corresponding discrete domain rational transfer function. There is an inherent problem with this discretization method using shift operator parameterization (discrete Z-domain). At fast sampling time, the discretized version of the continuous time operator or system should resemble that of the continuous time counterpart if the sampling theorem is satisfied. At very high sampling rate, the shift operator parameterized system fails to provide meaningful information due to its numerical ill conditioning. To overcome this problem, Delta operator parameterization for discretization is considered in this paper, where at fast sampling rate, the continuous time results can be obtained from the discrete time experiments and therefore a unified framework can be developed to get the discrete time results and continuous time results hand to hand. In this paper a new generating function is proposed to discretize the fractional order operator using the Gauss-Legendre 2-point quadrature rule. Additionally, the function has been expanded using the CFE in order to obtain rational approximation of the fractional order operator. The detailed mathematical formulations along with the simulation results in MATLAB, with different fractional order systems are considered, in order to prove the newness of this formulation for discretization of the FOS in complex Delta domain.

The spectral collocation method for solving a fractional integro-differential equation

<abstract><p>In this paper, we propose a high-precision numerical algorithm for a fractional integro-differential equation based on the shifted Legendre polynomials and the idea of Gauss-Legendre quadrature rule and spectral collocation method. The error analysis of this method is also given in detail. Some numerical examples are give to illustrate the exponential convergence of our method.</p></abstract>