Multistage Numerical Picard Iteration Methods for Nonlinear Volterra Integral Equations of the Second Kind

Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.

Numerical Solution of Generalized Abel’s Integral Equation by Variational Iteration Method

In this paper, a user friendly algorithm based on the variational iteration method (VIM) is proposed to solve singular integral equations with generalized Abel’s kernel. It is observed that an approximate solutions yn(x) converges to the exact solution irrespective of the initial choice y0 (x). Illustrative numerical examples are given to demonstrate the efficiency and simplicity of the method in solving these types of singular integral equations.

Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method

In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.

An Improved Algorithm for 3D Reconstruction Integration Based on Stripe Reflection Method

This paper introduces the basic principle of stripe reflection method and proposes an improved algorithm on the traditional Southwell gradient iterative integration algorithm. The algorithm adds a coefficient value with an attenuation factor to the compensation height value and the value of the attenuation factor is changed by the determination of the compensation height threshold. Through computer simulation, the fitting error of the reconstructed surface show that the RMS of the new method is one order of magnitude better than the traditional algorithm and the PV value of the high frequency part is about 1/15 of the traditional algorithm. It is proved that the improved algorithm can effectively improve the convergence and noise resistance of the iterative algorithm.

A Least-squares-based Iterative Method with Better Convergence for PF/OPF in Integrated Transmission and Distribution Networks

The limitations of the conventional master-slavesplitting(MSS)method,which is commonly applied to power flow and optimal power flow in integrated transmission and distribution(I-T&D)networks,are first analyzed.Considering that the MSS method suffers from a slow convergence rate or even divergence under some circumstances,a least-squares-based iterative(LSI)method is proposed.Compared with the MSS method,the LSI method modifies the iterative variables in each iteration by solving a least-squares problem with the information in previous iterations.A practical implementation and a parameter tuning strategy for the LSI method are discussed.Furthermore,a LSI-PF method is proposed to solve I-T&D power flow and a LSIheterogeneous decomposition(LSI-HGD)method is proposed to solve optimal power flow.Numerical experiments demonstrate that the proposed LSI-PF and LSI-HGD methods can achieve the same accuracy as the benchmark methods.Meanwhile,these LSI methods,with appropriate settings,significantly enhance the convergence and efficiency of conventional methods.Also,in some cases,where conventional methods diverge,these LSI methods can still converge.

ITERATIVE ALGORITHM FOR AXIALLY ACCELERATING STRINGS WITH INTEGRAL CONSTITUTIVE LAW

A numerical method is proposed to simulate the transverse vibrations of a viscoelastic moving string constituted by an integral law. In the numerical computation, the Galerkin method based on the Hermite functions is applied to discretize the state variables, and the Runge- Kutta method is applied to solve the resulting differential-integral equation system. A linear iterative process is designed to compute the integral terms at each time step, which makes the numerical method more efficient and accurate. As examples, nonlinear parametric vibrations of an axially moving viscoelastic string are analyzed.

Some Approximation in Cone Metric Space and Variational Iterative Method

In this paper, we give some new results of common fixed point theorems and coincidence point case for some iterative method. By using of variation iteration method and an effective modification of He’s variation iteration method discusses some integral and differential equations, we give out some new conclusion and more new examples.

Operator Equation and Application of Variation Iterative Method

In this paper, we study some semi-closed 1-set-contractive operators A and investigate the boundary conditions under which the topological degrees of 1-set contractive fields, deg (I-A, Ω, p) are equal to 1. Correspondingly, we can obtain some new fixed point theorems for 1-set-contractive operators which extend and improve many famous theorems such as the Leray-Schauder theorem, and operator equation, etc. Lemma 2.1 generalizes the famous theorem. The calculation of topological degrees and index are important things, which combine the existence of solution of for integration and differential equation and or approximation by iteration technique. So, we apply the effective modification of He’s variation iteration method to solve some nonlinear and linear equations are proceed to examine some a class of integral-differential equations, to illustrate the effectiveness and convenience of this method.

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.

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable;therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type;then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

DQM2022系列超高阶地球重力场模型构建及其精度评估

本文基于椭球谐分析,提出了超高阶地球重力场模型构建的局部积分改进迭代法和全球积分改进迭代法,解决了传统的局部积分改进法存在的局限性,有效提高了改进模型在中国地区的适用精度。选择EGM2008、EIGEN-6C4地球重力场模型作为初始模型,利用中国及周边地区最新的5′×5′实测格网平均重力异常数据,构建了DQM2022系列超高阶地球重力场模型,其完全阶次均为2190。采用地面实测重力异常、GNSS/水准、天文大地垂线偏差等数据对改进模型进行精度评估。结果显示:①相对于初始模型,改进模型表示中国地区重力场的精度显著提高,重力异常精度提高了2.4~2.8 mGal,高程异常精度提高了1.0~2.4 cm,垂线偏差精度提高了0.07″~0.15″;②在表示中国地区重力场时,基于EIGEN-6C4初始模型构建的改进模型精度最高,高程异常精度约为10.6 cm,垂线偏差精度约为2.1″。

车辆-轨道非线性耦合系统显隐式求解算法的收敛性分析

针对车辆-轨道非线性耦合系统的动力学方程联立求解过程,将车辆和轨道分别考虑为2个子系统,通过2个子系统之间的位移协调和轮轨非线性接触力耦合,再结合显隐式积分格式,提出车辆-轨道非线性耦合系统的分离迭代和分离同步两种求解方法,通过算例验证了两种方法的正确性,并对两种求解方法的收敛性进行了对比分析。计算结果表明:显隐式分离迭代法和显隐式分离同步法的最大有效时间步长分别是0.5、0.1 ms;在时间步长较大时,采用Aitken加速法可以起到增强显隐式分离迭代法的计算稳定性的作用,但Aitken加速法的作用会随着时间步长的减小而变小;相同时间步长下显隐式分离同步法的计算效率要比显隐式分离迭代法高,但可以通过选用较大的时间步长来提高显隐式分离迭代法的计算效率。

Fast and accurate adaptive collocation iteration method for orbit dynamic problems

For over half a century,numerical integration methods based on finite difference,such as the Runge-Kutta method and the Euler method,have been popular and widely used for solving orbit dynamic problems.In general,a small integration step size is always required to suppress the increase of the accumulated computation error,which leads to a relatively slow computation speed.Recently,a collocation iteration method,approximating the solutions of orbit dynamic problems iteratively,has been developed.This method achieves high computation accuracy with extremely large step size.Although efficient,the collocation iteration method suffers from two limitations:(A)the computational error limit of the approximate solution is not clear;(B)extensive trials and errors are always required in tuning parameters.To overcome these problems,the influence mechanism of how the dynamic problems and parameters affect the error limit of the collocation iteration method is explored.On this basis,a parameter adjustment method known as the“polishing method”is proposed to improve the computation speed.The method proposed is demonstrated in three typical orbit dynamic problems in aerospace engineering:a low Earth orbit propagation problem,a Molniya orbit propagation problem,and a geostationary orbit propagation problem.Numerical simulations show that the proposed polishing method is faster and more accurate than the finite-difference-based method and the most advanced collocation iteration method.

一类三阶积分边值问题的两个迭代正解

利用上下解结合单调迭代方法建立一类三阶积分边值问题两个迭代正解的存在性。通过建立相应线性积分边值问题的格林函数得到解的表达式,并讨论其格林函数性质。将该边值问题解的存在性问题转化为锥上算子的不动点问题,并建立该算子的全连续性。在不用假设上下解存在的情况下,通过构造上下解并应用单调迭代方法建立该边值问题正解的存在性,同时得到逼近正解的迭代格式。最后给出了结论的具体应用。

基于多区多求解器的金属-介质复合目标电磁散射特性分析

在导弹类金属-介质复合目标电磁散射特性求解过程中,采用常规迭代求解方法存在难以收敛以及内迭代边界积分区域重复求解的问题。针对该问题,在传统有限元边界积分区域分解法(finite element boundary integral domain decomposition method,FE-BI-DDM)的基础上,采用了更为灵活的多区多求解器的方法(multi domain multi solver method,MDMSM)。该方法对导弹类金属-介质复合目标中难以收敛的金属区域,使用快速直接求逆的方法求解,由于可以使用独立的网格模型进行电磁建模,避免了内迭代部分的模型重复建立过程,从而大幅减少了整体模型求解时间。实验结果表明:所提方法可以在相同计算精度的条件下,以不过多增加内存空间为前提,大幅缩短了导弹类目标的金属-介质复合模型的电磁求解时间。该方法为开展导弹类目标特性分析提供了一条可行的技术途径。

Response of train-bridge system under intensive seismic excitation by random vibration method

Earthquake is a kind of sudden and destructive random excitation in nature.It is significant to determine the probability distribution characteristics of the corresponding dynamic indicators to ensure the safety and the stability of structures when the intensive seismic excitation,the intensity of which is larger than 7,acts in train-bridge system.Firstly,the motion equations of a two-dimensional train-bridge system under the vertical random excitation of track irregularity and the vertical seismic acceleration are established,where the train subsystem is composed of 8 mutually independent vehicle elements with 48 degrees of freedom,while the single-span simple supported bridge subsystem is composed of 102D beam elements with 20 degrees of freedom on beam and 2 large mass degrees of freedom at the support.Secondly,Monte Carlo method and pseudo excitation method are adopted to analyze the statistical parameters of the system.The power spectrum density of random excitation is used to define a series of non-stationary pseudo excitation in pseudo excitation method and the trigonometric series of random vibration history samples in Monte Carlo method,respectively solved by precise integral method and Newmark-βmethod through the inter-system iterative procedure.Finally,the results are compared with the case under the weak seismic excitation,and show that the samples of vertical acceleration response of bridge and the offload factor of train obeys the normal distribution.In a high probability,the intensive earthquakes pose a greater threat to the safety and stability of bridges and trains than the weak ones.

Computation of the Eigenvalues of 3<i>D</i>“Charged” Integral Equations

The Rayleigh-Ritz and the inverse iteration methods are used in order to compute the eigenvalues of 3D Fredholm-Stieltjes integral equations, i.e. 3D Fredholm equations with respect to suitable Stieltjes-type measures. Some applications are shown, relevant to the problem of computing the eigenvalues of a body charged by a finite number of masses concentrated on points, curves or surfaces lying in.

基于虚拟激励法对高速铁路桥梁垂向共振响应的研究

轨道不平顺是常见的作用在列车-桥耦合系统的随机激励,其与列车移动轴重效应的确定性激励共同作用,使桥梁共振时的响应具有随机性。为克服单一轨道不平顺空间域样本给计算结果带来的不确定性和多样本计算的低效性,本文采用虚拟激励随机振动分析方法,构建二维车桥耦合系统空间状态方程,考虑轨道不平顺随机激励和确定性轴重激励,在列车以不同速度匀速行驶条件下,借助精细积分法通过全过程迭代过程对该状态方程进行精确高效求解,与传统的蒙特卡罗抽样下的纽马克贝塔法求解结果对比,并确定响应的概率分布模型。分析桥梁发生垂向共振时的时频响应,最终给列车安全过桥提出建议。

求解第二型Fredholm积分方程的迭代快速小波Petrov-Galerkin方法

给出求解带光滑核的第二型Fredholm积分方程的迭代快速小波Petrov-Galerkin方法,并分析该方法的收敛性以及计算复杂度,证明该方法可达到超收敛阶。

位场向下延拓的波数域迭代法及其收敛性

提出了位场向下延拓的波数域迭代法.对水平面上的位场观测值进行Fourier变换,得到其波谱.根据第一类Fredholm积分方程的空间域迭代解法,推导出计算向下延拓水平面上位场波谱的波数域迭代公式.在波数域中进行迭代,一直进行到相继两次迭代近似解的差值最大绝对值小于给定的精度,或迭代达到给定的最大迭代次数.对这种迭代近似解进行Fourier逆变换,得到向下延拓的位场.数值计算结果表明:与空间域迭代法比较,这种波数域迭代法简单、快速,并有同样好的向下延拓效果.本文还证明了这种迭代法是收敛的,并给出了它的收敛特性和滤波特性.