Approximate solution of Volterra-Fredholm integral equations using generalized barycentric rational interpolant

It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollable poles and low convergence order.In contrast with the classical rational interpolants,the generalized barycentric rational interpolants which depend linearly on the interpolated values,yield infinite smooth approximation with no poles in real numbers.In this paper,a numerical collocation approach,based on the generalized barycentric rational interpolation and Gaussian quadrature formula,was introduced to approximate the solution of Volterra-Fredholm integral equations.Three types of points in the solution domain are used as interpolation nodes.The obtained numerical results confirm that the barycentric rational interpolants are efficient tools for solving Volterra-Fredholm integral equations.Moreover,integral equations with Runge’s function as an exact solution,no oscillation occurrs in the obtained approximate solutions so that the Runge’s phenomenon is avoided.

Solving Navier-Stokes equation by mixed interpolation method

The operator splitting method is used to deal with the Navier-Stokes equation, in which the physical process described by the equation is decomposed into two processes： a diffusion process and a convection process; and the finite element equation is established. The velocity field in the element is described by the shape function of the isoparametric element with nine nodes and the pressure field is described by the interpolation function of the four nodes at the vertex of the isoparametric element with nine nodes. The subroutine of the element and the integrated finite element code are generated by the Finite Element Program Generator （FEPG） successfully. The numerical simulation about the incompressible viscous liquid flowing over a cylinder is carded out. The solution agrees with the experimental results very well.

Multi-quadric Equations Interpolation and Its Applications to the Establishment of Crustal Movement Speed Field

In this paper,multi_quadric equations interpolation is used to establish a widely covered and valuable speed field model,with which the crustal movement image is obtained.

A numerical method for one-dimensional nonlinear sine-Gordon equation using multiquadric quasi-interpolation

In this paper, we use a univariate multiquadric quasi-interpolation scheme to solve the one-dimensional nonlinear sine-Gordon equation that is related to many physical phenomena. We obtain a numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative and a difference scheme to approximate the temporal derivative. The advantage of the obtained scheme is that the algorithm is very simple so that it is very easy to implement. The results of numerical experiments are presented and compared with analytical solutions to confirm the good accuracy of the presented scheme.

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.

Numerical Solution of Freholm-Volterra Integral Equations by Using Scaling Function Interpolation Method

Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods. In this article, we use scaling function interpolation method to solve Volterra integral equations of the first kind, and Fredholm-Volterra integral equations. Moreover, we prove convergence theorem for the numerical solution of Volterra integral equations and Freholm-Volterra integral equations. We also present three examples of solving Volterra integral equation and one example of solving Fredholm-Volterra integral equation. Comparisons of the results with other methods are included in the examples.

A Meshless Collocation Method with Barycentric Lagrange Interpolation for Solving the Helmholtz Equation

In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.

Numerical Solution of Euler-Bernoulli Beam Equation by Using Barycentric Lagrange Interpolation Collocation Method

Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough high. In this paper, we will illustrate the high-precision numerical method to solve Euler-Bernoulli beam equation. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate new algorithm has the following advantages: small computational work, fast convergence speed and high precision.

LaNets:Hybrid Lagrange Neural Networks for Solving Partial Differential Equations

We propose new hybrid Lagrange neural networks called LaNets to predict the numerical solutions of partial differential equations.That is,we embed Lagrange interpolation and small sample learning into deep neural network frameworks.Concretely,we first perform Lagrange interpolation in front of the deep feedforward neural network.The Lagrange basis function has a neat structure and a strong expression ability,which is suitable to be a preprocessing tool for pre-fitting and feature extraction.Second,we introduce small sample learning into training,which is beneficial to guide themodel to be corrected quickly.Taking advantages of the theoretical support of traditional numerical method and the efficient allocation of modern machine learning,LaNets achieve higher predictive accuracy compared to the state-of-the-artwork.The stability and accuracy of the proposed algorithmare demonstrated through a series of classical numerical examples,including one-dimensional Burgers equation,onedimensional carburizing diffusion equations,two-dimensional Helmholtz equation and two-dimensional Burgers equation.Experimental results validate the robustness,effectiveness and flexibility of the proposed algorithm.

Two Families of Multipoint Root-Solvers Using Inverse Interpolation with Memory

In this paper, a general family of derivative-free n + 1-point iterative methods using n + 1 evaluations of the function and a general family of n-point iterative methods using n evaluations of the function and only one evaluation of its derivative are constructed by the inverse interpolation with the memory on the previous step for solving the simple root of a nonlinear equation. The order and order of convergence of them are proved respectively. Finally, the proposed methods and the basins of attraction are demonstrated by the numerical examples.

Research on Chaos of Nonlinear Singular Integral Equation

In this paper, one class of nonlinear singular integral equation is discussed through Lagrange interpolation method. We research the connections between numerical solutions of the equations and chaos in the process of solving by iterative method.

矿井通风参数缺失数据插补方法

矿井智能通风系统对矿山智能化建设至关重要。为解决矿井通风参数在实际测量时,因为巷道不具备测试条件、仪器信号受到干扰、巷道断面风速不均一、人工操作不当等制约性因素,造成的矿井通风参数数据缺失问题,提出了1种基于随机森林−链式方程多重插补法的矿井通风参数缺失数据插补方法。采用链式方程多重插补法,通过迭代对每个缺失的属性值产生n个插补值,从而产生n个完整数据集,对n个完整数据集进行分析优化得到1个最终的完整数据集。为了提高缺失值插补精度,合理考虑了矿井通风参数缺失数据的不确定性对分析过程的影响,在随机森林的预测任务中,结合预测均值匹配模型对缺失数据进行插补。以潞新二矿为实验对象,利用智能矿井通风仿真系统IMVS对潞新二矿矿井通风参数原始数据集进行数据预处理,得到完整、准确的矿井通风参数完整数据集,对完整数据集分别进行了不同缺失属性、不同数据缺失率、不同迭代次数的对比试验。以多种模型评价指标对模型有效性进行评估。结果表明:基于随机森林的链式方程多重插补模型插补形成的完整数据集与原始数据集具有很好的相似性;对不同缺失列进行插补实验的结果显示插补模型可以轻松处理混合类型的数据,自主学习参数之间的相关性从而降低了插补复杂性;迭代后形成的n个数据集通过分析合并成一个最终数据集,提高了插补准确率;对初始插补后的完整数据集进行不同迭代次数的试验,发现迭代超过一定次数后,数据相关性一定会收敛。

基于二元三次B样条拟插值的反应-扩散方程数值解

反应-扩散方程在科学和工程的许多分支中有着重要的应用,对此类方程数值解的研究具有重要意义.鉴于计算域的复杂形状、大量的自由度等导致计算非常困难,提出张量积型二元三次B样条法求解一类分数阶反应-扩散方程和交叉反应扩散系统,首先计算得出二元三次B样条拟插值的矩阵表达式,然后利用Matlab进行数值模拟,最后将数值模拟解与精确解进行对比.研究表明,当变量t的迭代次数较低时,所提方法行之有效.

带色散的四阶抛物型方程的紧致差分格式

本研究提出一种有效求解带色散四阶抛物型方程的四阶紧致差分格式。对该方程的空间变量用四阶紧致差分格式进行离散,对离散之后得到的常微分方程组用三次Hermite插值法进行求解,得到一种空间和时间方向上都具有四阶精度的数值格式,并用傅里叶方法证明了该格式的无条件稳定性。数值实验中给出三种类型的算例,并将本研究格式与Crank-Nicolson格式进行数值比较,证明了本研究格式的有效性。结果表明,本研究格式对求解带色散的四阶抛物型方程具有很好的实用性。

高校智能电表缺失数据修复方法

高校运行数据在采集、传输、存储过程中往往会产生数据缺失。对此,提出一种基于改进长短期记忆神经网络-链式方程多重插补法的缺失数据修复方法。采用链式方程多重插补法,通过迭代对每个缺失的属性值产生多个填补值,从而产生多个完整数据集,并进行分析优化得到一个最终的完整数据集。为提高缺失值修复精度,在长短期记忆神经网络的预测任务中,采用麻雀搜索算法进行超参数寻优,并结合均值匹配模型对缺失数据进行修复。使用北方某高校2019年数据进行验证,通过无自然缺失算例和自然缺失算例对提出方法进行评估,结果表明,在无自然缺失算例中,整体归因误差为0.106,较其他模型至少降低29.3%,验证了方法的有效性;对11.8%自然缺失率下的数据进行填补,经提出的方法填补之后的数据有效提高了高校后续运行数据的预测精度,间接验证了缺失数据填补的有效性。

基于双调和插值的锥束CT金属伪影校正算法

在计算机断层扫描(CT)中,金属植入物会引入严重的伪影,导致图像质量降低影响诊断价值。为了对锥束CT中的金属伪影进行校正,提出了一种基于双调和方程的金属伪影校正算法。首先,对含金属伪影的重建图像进行双边滤波和金属阈值分割,获得金属和非金属图像;然后,对二者进行正向投影,获得金属投影区域和先验投影图像;接下来,利用先验投影图像对原始投影进行归一化,并对金属区域进行双调和方程插值修复,获得修复的投影数据;最后,对修复的投影数据去归一化,并利用FDK算法进行重建,再与金属图像融合获得最终的校正图像。为了验证该算法的性能,采用实际拍摄的数据进行金属伪影校正实验。结果表明,与常用的线性插值校正算法和归一化校正算法相比,所提算法ROI区域内图像的均方根误差分别减少了22%和8%,有效地抑制了金属伪影,优于常用的去金属伪影算法。

Fredholm积分-微分方程的高精度数值方法研究

研究积分项包含未知函数导数的Fredholm积分-微分方程的重心插值配点法.首先,利用重心Lagrange插值配点法和重心有理插值配点法构造Fredholm积分-微分方程的数值格式.其次,分别选取等距节点和第二类Chebyshev节点进行数值计算,并对两种插值法在不同节点下的精度进行比较.数值算例结果表明,两种插值配点法都可以得到较高的计算精度,但一般情况下为了得到高精度的数值解,优先采用重心Lagrange插值配点法在第二类Chebyshev节点上计算.

基于特征矩阵分区等值和自适应插值切换的有源配电网多速率并行仿真方法

现代有源配电网呈现复杂多变、多时间尺度的特征,采用单一仿真速率难以同时兼顾计算效率与精度,对此该文提出基于特征矩阵分区等值和自适应插值切换的有源配电网多速率并行仿真方法。首先,构建系统状态空间矩阵,并基于状态矩阵特征值的量级将系统拆分为多个动态特性差异的子系统,通过将系统的状态矩阵进行分块等效,实现子区域在小步长时刻的独立仿真解算与在大步长时刻的等效补偿,在保证仿真精度的基础上大幅降低了系统仿真规模;然后,针对各区域间数据交互不同步问题,通过倍率步长相量法分析线性插值误差产生原理及主要影响因素,并提出基于二阶泰勒展开式的多步自适应插值仿真算法,可以根据系统频率与仿真步长需求调节相关插值参数,显著降低插值误差;最后,采用改进型IEEE 123节点配电网算例,通过分区等值/插值计算与仿真结果的对比,验证了所提多速率并行仿真方法的可行性与有效性。

高速发射装置柔性传动系统非线性运动研究

为满足高速发射装置的使用要求,针对弹链非线性运动导轨进行设计分析和优化,以确保弹体顺利传送至击发位置。导轨的设计优化是确保高射速、高精度发射系统供弹连续性和可靠性的关键。通过建立高速发射装置导轨曲线方程,并分析导轨不同位置的曲线特性,针对其设计中存在的柔性冲击,运用样条插值对导轨曲线进行优化设计,并对优化后的导轨进行仿真验证,验证结果满足发射装置的使用要求。文中旨在为改善发射装置非线性运动导轨动力学特性和提高导轨使用寿命提供理论依据。

变系数Volterra型积分微分方程的2种Legendre谱Galerkin数值积分方法

为了进一步提高求解Volterra型积分微分的数值精度,针对一种变系数Volterra型积分微分方程,提出了2种Legendre谱Galerkin数值积分法。采用Galerkin Legendre数值积分对Volterra型积分微分方程的积分项进行预处理,对其构造Legendre tau格式,同时用Chebyshev-Gauss-Lobatto配置点对变系数和积分项部分进行计算,并通过对方程的定义区间进行分解,提出了一种多区间Legendre谱Galerkin数值积分法。该方法的格式对于奇数阶模型具有对称结构。此外,通过引入Volterra型积分微分方程的最小二乘函数,构造了Legendre谱Galerkin最小二乘数值积分法。该方法对应的代数方程系数矩阵是对称正定的。数值算例验证了这2种Legendre谱Galerkin数值积分方法的高阶精度和有效性。