Some quadrature-based versions of the generalized Newton method for solving nonsmooth equations

In this paper, modifications of a generalized Newton method based on some rules of quadrature are studied. The methods considered are Newton-like iterative schemes for numerical solving systems of nonsmooth equations. Some mild conditions are given that ensure superlinear convergence to a solution. Moreover, a parameterized version of the midpoint version is presented. Finally, results of numerical tests are established.     

一类含中介值定积分等式证明题的构造

利用多项式插值理论,结合数值求积公式代数精度的概念,给出了一种构造和证明一类含中介值定积分等式证明题的新方法,并给出多个应用实例.     

Saturation-free numerical scheme for computing the flow past a lattice of airfoils and the determination of separation points in a viscous fluid

A numerical method for computing the potential flow past a lattice of airfoils is described. The problem is reduced to a linear integrodifferential equation on the lattice contour, which is then approximated by a linear system of equations with the help of specially derived quadrature formulas. The quadrature formulas exhibit exponential convergence in the number of points on an airfoil and have a simple analytical form. Due to its fast convergence and high accuracy, the method can be used to directly optimize the airfoils as based on any given integral characteristics. The shear stress distribution and the separation points are determined from the velocity distribution at the airfoil boundary calculated by solving the boundary layer equations. The method proposed is free of laborious grid generation procedures and does not involve difficulties associated with numerical viscosity at high Reynolds numbers.     

Numerical Solution of non-Homogeneous Semi-Markov Processes in Transient Case*

In this article a numerical solution for the evolution equation of a continuous time non-homogeneous semi-Markov process (NHSMP) is obtained using a quadrature method. The paper, after a short introduction to continuous time NHSMP, presents the numerical solution of the process evolution equation with a general quadrature method. Furthermore, the paper gives results that justify this approach, proving that the numerical solution tends to the evolution equation of the continuous time NHSMP. Moreover, the formulae related to some specific quadrature methods are given and a method for obtaining the discrete time NHSMP by applying a very particular quadrature formula for the discretization is shown. In this way the relation between the continuous and discrete time NHSMP is proved. Then, the problem of obtaining the continuous time NHSMP from the discrete one is considered. This problem is solved showing that the discrete process converges in law to the continuous one if the discretized time interval tends to zero. In addition, the discrete time NHSMP in matrix form is presented, and the fact that the solution to this process always exists is proved. Finally, an algorithm for solving the discrete time NHSMP is given. To illustrate the use of this algorithm for a discrete NHSMP, an example in the area of finance is presented.     

分布阶扩散—波动方程的有限元解的误差估计

本文考虑了分布阶时间分数阶扩散波动方程,其中时间分数阶导数是在Caputo意义上定义的,其阶次$\alpha,\beta$分别属于（0,1）和（1,2）.文中提出了在计算上行之有效的数值方法来模拟分布阶时间分数阶扩散波动方程.在时间上,通过中点求积公式把分布阶项转换为多项的时间分数阶导数项,并且利用$L1$和$L2$公式来近似Caputo分数阶导数;空间上使用Galerkin有限元方法进行离散.给出了基于$H^1$范数的有限元解的稳定性和误差估计的详细证明,最后的数值算例结果说明了理论分析的正确性以及有效性.     

三步五阶迭代方法解非线性方程组

本文根据求积公式, 给出了三种求解非线性方程组的迭代方法, 并证明了所提出的三步迭代方法具有五阶收敛性. 最后给出了四个数值实例, 将本文的实验结果与现有的几种迭代方法的实验结果作了比较分析, 表明本文所提出的方法具有明显的优越性.     

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L ² and the H ¹ norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.     

Numerical Treatment of Homogeneous Semi-Markov Processes in Transient Case–a Straightforward Approach

This paper presents the numerical solution of the process evolution equation of a homogeneous semi-Markov process (HSMP) with a general quadrature method. Furthermore, results that justify this approach proving that the numerical solution tends to the evolution equation of the continuous time HSMP are given. The results obtained generalize classical results on integral equation numerical solutions applying them to particular kinds of integral equation systems. A method for obtaining the discrete time HSMP is shown by applying a very particular quadrature formula for the discretization. Following that, the problem of obtaining the continuous time HSMP from the discrete one is considered. In addition, the discrete time HSMP in matrix form is presented and the fact that the solution of the evolution equation of this process always exists is proved. Afterwards, an algorithm for solving the discrete time HSMP is given. Finally, a simple application of the HSMP is given for a real data social security example.     

Szegő–Lobatto quadrature rules

Gauss-type quadrature rules with one or two prescribed nodes are well known and are commonly referred to as Gauss–Radau and Gauss–Lobatto quadrature rules, respectively. Efficient algorithms are available for their computation. Szeg? quadrature rules are analogs of Gauss quadrature rules for the integration of periodic functions; they integrate exactly trigonometric polynomials of as high degree as possible. Szeg? quadrature rules have a free parameter, which can be used to prescribe one node. This paper discusses an analog of Gauss–Lobatto rules, i.e., Szeg? quadrature rules with two prescribed nodes. We refer to these rules as Szeg?–Lobatto rules. Their properties as well as numerical methods for their computation are discussed.     

Solving multivariable mathematical models by the quadrature and cubature methods

The utilization and generalization of quadrature and cubature approximations for numerical solution of mathematical models of multivariable transport processes involving integral, differential, and integro-differential operators, and for numerical interpolation and extrapolation, are presented. The methodology for determination of the quadrature and cubature weights for composite operators is developed to accommodate for general functional representations. Application of these methods is demonstrated by solving two-dimensional steady-state and one-dimensional transient-state problems. The solutions are compared with exact-analytical solutions to evaluate the performance of these methods. It is demonstrated that the quadrature and cubature approximations are simple and universal; i.e., the same formula is applicable irrespective of the order of accuracy of the numerical approximation, the type of linear operator, and the number of temporal and/or spatial variables. Since the quadrature and cubature methods can produce solutions with sufficient accuracy even when using fewer discrete points, both the programming task and computational effort are reduced considerably. Therefore, the quadrature and cubature methods appear to be very practical in solving the mathematical models of a variety of transport processes. © 1994 John Wiley & Sons, Inc.     

两类五阶解非线性方程组的迭代算法

本文首先根据Runge-Kutta方法的思想,结合Newton迭代法,提出了一类带参数的解非线性方程组F(x)=0的迭代算法,然后基于解非线性方程f(x)=0的King算法,给出第二类解非线性方程组的迭代算法,收敛性分析表明这两类算法都是五阶收敛的.其次给出了本文两类算法的效率指数,以及一些已知算法的效率指数,并且将本文算法的效率指数与其它方法进行详细的比较,通过效率比率R_(i,j)可知本文算法具有较高的计算效率.最后给出了四个数值实例,将本文两类算法与现有的几种算法进行比较,实验结果说明本文算法收敛速度快,迭代次数少,有明显的优势.     

ON QUADRATURE FORMULAE FOR SINGULAR INTEGRALS OF ARBITRARY ORDER

Some quadrature formulae for the numerical evaluation of singular integrals of arbitrary order are established and both the estimate of remainder and the convergence of each quadrature formula derived here are also given.     

刚性多滞量积分微分方程的Runge—Kutta方法

本文研究了求解刚性多滞量积分微分方程的Runge-Kutta方法的非线性稳定性和计算有效性.经典Runge—Kutta方法连同复合求积公式和Pouzet求积公式被改造用于求解一类刚性多滞量Volterra型积分微分方程.其分析导出了:在适当条件下,扩展的Runge-Kutta方法是渐近稳定和整体稳定的.此外,数值试验表明所给出的方法是高度有效的.     

B-Spline collocation method for linear and nonlinear Fredholm and Volterra integro-differential equations

A numerical method based on quintic B-spline has been developed to solve the linear and nonlinear Fredholm and Volterra integro-differential equations up to order 4. The solution and its derivatives are collocated by quintic B-spline and then the integral equation is approximated by the 4-points Gauss–Turán quadrature formula with respect to the weight function Legendre. The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method which shows that our method can be applied for large values of N. The results are compared with the results obtained by other methods which show that our method is accurate.     

A coupled finite element-differential quadrature element method and its accuracy for moving load problem

In this paper a mixed method, which combines the finite element method and the differential quadrature element method (DQEM), is presented for solving the time dependent problems. In this study, the finite element method is first used to discretize the spatial domain. The DQEM is then employed as a step-by-step DQM in time domain to solve the resulting initial value problem. The resulting algebraic equations can be solved by either direct or iterative methods. Two general formulations using the DQM are also presented for solving a system of linear second-order ordinary differential equations in time. The application of the formulation is then shown by solving a sample moving load problem. Numerical results show that the present mixed method is very efficient and reliable.     

Generalization of differential quadrature discretization

The differential quadrature (DQ) is generalized. Various methods for generating the weighting coefficients are developed. The design of a grid model is flexible. Weighting coefficients for general multi-coordinate grid models with arbitrary configurations can also be calculated. The calculation of weighting coefficients is easy. Sample numerical procedures for constructing one-coordinate, two-coordinate and arbitrary finite-coordinate generic differential quadrature models are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

解带有Hilbert核的奇异积分方程的高精度组合算法

该文给出了用求积法解带Hilber核的奇异积分方程的高精度组合算法.把网格点分成互不相交的子集合, 在子集合上并行求解离散方程组, 再利用组合算法求得全局网格点的逼近.如果积分方程的系数属于B_δ, 则求积法的精度可达O(e^-nδ). 此外, 使用组合算法不仅能得到更高的精度阶, 而且能够得到后验误差估计. 数值算例的结果表明组合算法是极其有效的.     

Block by block method for the systems of nonlinear Volterra integral equations

The approach given in this paper leads to numerical methods for solving system of Volterra integral equations which avoid the need for special starting procedures. The method has also the advantages of simplicity of application and at least four order of convergence which is easy to achieve. Also, at each step we get four unknowns simultaneously. A convergence theorem is proved for the described method. Finally numerical examples presented to certify convergence and accuracy of the method.     

Numerical solution of the Orr–Sommerfeld equation using the viscous Green function and split-Gaussian quadrature

We continue our study of the construction of numerical methods for solving two-point boundary value problems using Green functions, building on the successful use of split-Gauss-type quadrature schemes. Here we adapt the method for eigenvalue problems, in particular the Orr–Sommerfeld equation of hydrodynamic stability theory. Use of the Green function for the viscous part of the problem reduces the fourth-order ordinary differential equation to an integro-differential equation which we then discretize using the split-Gaussian quadrature and product integration approach of our earlier work along with pseudospectral differentiation matrices for the remaining differential operators. As the latter are only second-order the resulting discrete equations are much more stable than those obtained from the original differential equation. This permits us to obtain results for the standard test problem (plane Poiseuille flow at unit streamwise wavenumber and Reynolds number 10 000) that we believe are the most accurate to date.     

Jacobi-Picard iteration method for the numerical solution of nonlinear initial value problems

In this paper, an effective numerical iterative method for solving nonlinear initial value problems (IVPs) is presented. The proposed iterative scheme, called the Jacobi-Picard iteration (JPI) method, is based on the Picard iteration technique, orthogonal shifted Jacobi polynomials, and shifted Jacobi-Gauss quadrature formula. In comparison with traditional methods, the JPI method uses an iterative formula for updating next step approximations and calculating integrals of the shifted Jacobi polynomials are performed via an exact relation. Also, a vector-matrix form of the JPI method is provided in details which reduce the CPU time. The performance of the presented method has been investigated by solving several nonlinear IVPs. Numerical results show the efficiency and the accuracy of the proposed iterative method.