解奇异非光滑方程组的牛顿法和不精确牛顿法

本文主要解决奇异非光滑方程组的解法。应用一种新的次微分的外逆，我们提出了牛顿法和不精确牛顿法，它们的收敛性同时也得到了证明。这种方法能更容易在一引起实际应用中实现。这种方法可以看作是已存在的解非光滑方程组的方法的延伸。     

一个三阶牛顿变形方法

基于反函数建立的积分方程,结合Simpson公式,给出了一个非线性方程求根的新方法,即为牛顿变形方法.证明了它至少三次收敛到单根,与牛顿法相比,提高了收敛阶和效率指数.文末给出数值试验,且与牛顿法和同类型牛顿变形法做了比较.结果表明方法具有较好的优越性,它丰富了非线性方程求根的方法.     

A fourth-order approximate projection method for the incompressible Navier–Stokes equations on locally-refined periodic domains

In this follow-up of our previous work [30], the author proposes a high-order semi-implicit method for numerically solving the incompressible Navier–Stokes equations on locally-refined periodic domains. Fourth-order finite-volume stencils are employed for spatially discretizing various operators in the context of structured adaptive mesh refinement (AMR). Time integration adopts a fourth-order, semi-implicit, additive Runge–Kutta method to treat the non-stiff convection term explicitly and the stiff diffusion term implicitly. The divergence-free condition is fulfilled by an approximate projection operator. Altogether, these components yield a simple algorithm for simulating incompressible viscous flows on periodic domains with fourth-order accuracies both in time and in space. Results of numerical tests show that the proposed method is superior to previous second-order methods in terms of accuracy and efficiency. A major contribution of this work is the analysis of a fourth-order approximate projection operator.     

Numerical comparison of methods for solving linear differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrates that the new methods are quite accurate and readily implemented.     

Partitioned quasi-Newton methods for sparse nonlinear equations

In this paper, we present two partitioned quasi-Newton methods for solving partially separable nonlinear equations. When the Jacobian is not available, we propose a partitioned Broyden’s rank one method and show that the full step partitioned Broyden’s rank one method is locally and superlinearly convergent. By using a well-defined derivative-free line search, we globalize the method and establish its global and superlinear convergence. In the case where the Jacobian is available, we propose a partitioned adjoint Broyden method and show its global and superlinear convergence. We also present some preliminary numerical results. The results show that the two partitioned quasi-Newton methods are effective and competitive for solving large-scale partially separable nonlinear equations.     

A New defy for iteration methods

The work presents an adaptation of iteration method for solving a class of thirst order partial nonlinear differential equation with mixed derivatives.The class of partial differential equations present here is not solvable with neither the method of Green function, the most usual iteration methods for instance variational iteration method, homotopy perturbation method and Adomian decomposition method, nor integral transform for instance Laplace,Sumudu, Fourier and Mellin transform. We presented the stability and convergence of the used method for solving this class of nonlinear chaotic equations.Using the proposed method, we obtained exact solutions to this kind of equations.     

Third-order iterative methods free from second derivatives for nonlinear equation

In this paper, we suggest and analyze a new two-step iterative method for solving nonlinear equations, which is called the modified Householder method without second derivatives for nonlinear equation. We also prove that the modified method has cubic convergence. Several examples are given to illustrate the efficiency and the performance of the new method. New method can be considered as an alternative to the present cubic convergent methods for solving nonlinear equations.     

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.     

Modified Jarratt method for computing multiple roots

In this paper, we present a fourth order method for computing multiple roots of nonlinear equations. The method is based on Jarratt scheme for simple roots [P. Jarratt, Some efficient fourth order multipoint methods for solving equations, BIT 9 (1969) 119-124]. The method is optimal, since it requires three evaluations per step, namely one evaluation of function and two evaluations of first derivative. The efficacy is tested on a number of relevant numerical problems. It is observed that the present scheme is competitive with other similar robust methods.     

Preconditioned tensor format conjugate gradient squared and biconjugate gradient stabilized methods for solving stein tensor equations

This article is concerned with solving the high order Stein tensor equation arising in control theory. The conjugate gradient squared (CGS) method and the biconjugate gradient stabilized (BiCGSTAB) method are attractive methods for solving linear systems. Compared with the large-scale matrix equation, the equivalent tensor equation needs less storage space and computational costs. Therefore, we present the tensor formats of CGS and BiCGSTAB methods for solving high order Stein tensor equations. Moreover, a nearest Kronecker product preconditioner is given and the preconditioned tensor format methods are studied. Finally, the feasibility and effectiveness of the new methods are verified by some numerical examples.     

Shifted skew-symmetric iteration methods for nonsymmetric linear complementarity problems

We present a shifted skew-symmetric iteration method for solving the nonsymmetric positive definite or positive semidefinite linear complementarity problems. This method is based on the symmetric and skew-symmetric splitting of the system matrix, which has been adopted to establish efficient splitting iteration methods for solving the nonsymmetric systems of linear equations. Global convergence of the method is proved, and the corresponding inexact splitting iteration scheme is established and analyzed in detail. Numerical results show that the new methods are feasible and effective for solving large sparse and nonsymmetric linear complementarity problems.     

解可分离结构变分不等式的一种新的交替方向法

交替方向法是求解可分离结构变分不等式问题的经典方法之一, 它将一个大型的变分不等式问题分解成若干个小规模的变分不等式问题进行迭代求解. 但每步迭代过程中求解的子问题仍然摆脱不了求解变分不等式子问题的瓶颈. 从数值计算上来说, 求解一个变分不等式并不是一件容易的事情.因此, 本文提出一种新的交替方向法, 每步迭代只需要求解一个变分不等式子问题和一个强单调的非线性方程组子问题. 相对变分不等式问题而言, 我们更容易、且有更多的有效算法求解一个非线性方程组问题. 在与经典的交替方向法相同的假设条件下, 我们证明了新算法的全局收敛性. 进一步的数值试验也验证了新算法的有效性.     

Newton and quasi-Newton methods for equations of smooth compositions of semismooth functions

The Newton method and the quasi-Newton method for solving equations of smooth compositions of semismooth functions are proposed. The Q-superlinear convergence of the Newton method and the Q-linear convergence of the quasi-Newton method are proved. The present methods can be more easily implemented than previous ones for this class of nonsmooth equations.     

Iterative methods for nonlinear systems associated with finite difference approach in stochastic differential equations

We present some iterative methods of different convergence orders for solving systems of nonlinear equations. Their computational complexities are studies. Then, we introduce the method of finite difference for solving stochastic differential equations of Itô-type. Subsequently, our multi-step iterative schemes are employed in this procedure. Several experiments are finally taken into account to show that the presented approach and methods work well.     

Various Newton-type iterative methods for solving nonlinear equations

The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterative methods for solving nonlinear equations. The ninth order convergent Newton-type iterative method is made derivative free to obtain seventh-order convergent Newton-type iterative method. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterative methods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.     

(2,q)阶分数差分方程的解

首次提出了一种分数阶差分,分数阶和分以及分数阶差分方程的定义,并给出(2,q)阶常系数分数阶差分方程的具体解法.     

Numerical algorithm for solving multi-pantograph delay equations on the half-line using Jacobi rational functions with convergence analysis

A new spectral Jacobi rational-Gauss collocation (JRC) method is proposed for solving the multi-pantograph delay differential equations on the half-line. The method is based on Jacobi rational functions and Gauss quadrature integration formula. The main idea for obtaining a semi-analytical solution for these equations is essentially developed by reducing the pantograph equations with their initial conditions to systems of algebraic equations in the unknown expansion coefficients. The convergence analysis of the method is analyzed. The method possesses the spectral accuracy. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Indeed, the present method is compared favorably with other methods.     

Stable strong order 1.0 schemes for solving stochastic ordinary differential equations

The Balanced method was introduced as a class of quasi-implicit methods, based upon the Euler-Maruyama scheme, for solving stiff stochastic differential equations. We extend the Balanced method to introduce a class of stable strong order 1.0 numerical schemes for solving stochastic ordinary differential equations. We derive convergence results for this class of numerical schemes. We illustrate the asymptotic stability of this class of schemes is illustrated and is compared with contemporary schemes of strong order 1.0. We present some evidence on parametric selection with respect to minimising the error convergence terms. Furthermore we provide a convergence result for general Balanced style schemes of higher orders.     

牛顿-正则化方法与一类差分方程反问题的求解

在用牛顿迭代法求解非线性算子方程时,总要求非线性算子的导算子是有界可逆的,即线性化方程是适定的.但在实际数值计算中.即使满足这个条件,也可能出现数值不稳定的现象.为了克服这个困难,[1]将牛顿法与求解线性不适定问题的BG方法(平均核方法)结合起来,在每一步迭代中利用BG方法稳定求解.考虑到Tikhonov的正则化方