利用蒙特卡罗方法求解数值积分

本文介绍了蒙特卡罗方法的主要思想和理论基础,以一维定积分与二维定积分为例,应用蒙特卡罗方法借助R语言模拟计算定积分的值,并给出结果的相对误差与样本量的关系.     

利用定积分求解一阶常微分方程特解的方法

介绍利用定积分求解常见一阶常微分方程特解的基本方法,并借助实例进行具体计算.     

关于用定积分求解实际问题的方法讨论

本文给出了将部分重积分问题转化为定积分求解的一些推论，讨论了应用中运用“微元法”解决实际问题的方法。     

求解分数阶延迟微分方程的卷积Runge-Kutta方法

本文利用强A-稳定Runge-Kutta方法求解一类非线性分数阶延迟微分方程初值问题,并给出了算法的稳定性和误差分析.数值算例验证算法的有效性及其相关理论结果.     

一种新的正则化方法求解热传导方程的侧边值问题

The seriously ill-posed sideways heat equations were considered in the quarter plane. The classical quasi-reversibility method was applied to acquire an approximate but non-regularized solution to the problem. Interestingly, a regularization solution to the sideways heat equation was obtained through modification of the denominator of the solution. Then, a new regularization method was proposed, and the Hölder-type error estimates under a priori and a posteriori parameter choice rules were proved, respectively. Numerical experiments show the feasibility and effectiveness of the proposed method. © Editorial Office of Applied Mathematics and Mechanics.     

一类新的求解非线性方程的七阶方法

利用权函数法给出了一类求解非线性方程单根的七阶收敛的方法.每步迭代需要计算三个函数值和一个导数值,因此方法的效率指数为1.627.数值试验给出了该方法与牛顿法及同类方法的比较,显示了该方法的优越性.最后指出Kou等人给出的七阶方法是方法的特例.     

第一类算子方程的一种新的迭代正则化方法

提出了一种新的解第一类算子方程的迭代正则化方法,与通常的迭代正则化方法相比,提高了j次迭代正则解的渐近阶估计.同时,给出了后验正则化参数的选择.     

基于Monte-Carlo随机有限元方法的随机边界条件下自然对流换热不确定性研究

为分析边界条件不确定性对方腔内自然对流换热的影响,发展了一种求解随机边界条件下自然对流换热不确定性传播的Monte-Carlo随机有限元方法.通过对输入参数场随机边界条件进行Karhunen-Loeve展开及基于Latin(拉丁)抽样法生成边界条件随机样本,数值计算了不同边界条件随机样本下方腔内自然对流换热流场与温度场,并用采样统计方法计算了随机输出场的平均值与标准偏差.根据计算框架编写了求解随机边界条件下方腔内自然对流换热不确定性的MATLAB随机有限元程序,分析了随机边界条件相关长度与方差对自然对流不确定性的影响.结果表明:平均温度场及流场与确定性温度场及流场分布基本相同;随机边界条件下Nu数概率分布基本呈现正态分布,平均Nu数随着相关长度和方差增加而增大;方差对自然对流换热的影响强于相关长度的影响.     

基于正态分布求解两类反常积分

对两类反常积分,进行巧妙处理,构造成正态分布函数形式,利用正态分布函数的正则性进行求解.     

蒙特卡罗方法计算定积分的进一步讨论

介绍了蒙特卡罗方法计算定积分的原理和方法.给出了用蒙特卡罗方法计算定积分的一个简单证明,从而揭示了蒙特卡罗方法和定积分定义间的内在联系.针对蒙特卡罗方法收敛慢的特点,提出将蒙特卡罗方法与相应的数值计算方法相结合,提高计算结果的精度.此外,将蒙特卡罗方法推广到反常积分上去.     

波动方程初值问题的解在两类积分中的应用

利用波动方程初值问题解的特点给出了圆域上一类反常二重积分和球面上第一类曲面积分的微分算子级数公式解和定积分公式解.通过举例说明了该方法相对于常规解法的简便实用性.     

定积分计算方法及其数值试验

以定积分概念所蕴含的数学思想和哲学思想为指导,给出定积分近似计算的两种方法,即定义法和蒙特卡罗方法,并借助Matlab软件编程对其加以验证,数值试验结果显示所给方法可行且有效．     

Newton Methods for Solving Two Classes of Nonsmooth Equations

The paper is devoted to two systems of nonsmooth equations. One is the system of equations of max-type functions and the other is the system of equations of smooth compositions of max-type functions. The Newton and approximate Newton methods for these two systems are proposed. The Q-superlinear convergence of the Newton methods and the Q-linear convergence of the approximate Newton methods are established. The present methods can be more easily implemented than the previous ones, since they do not require an element of Clarke generalized Jacobian, of B-differential, or of b-differential, at each iteration point.     

正倒向随机微分方程组的数值解法

1990年,Pardoux和Peng(彭实戈)解决了非线性倒向随机微分方程(backward stochastic differential equation,BSDE)解的存在唯一性问题,从而建立了正倒向随机微分方程组(forward backward stochastic differential equations,FBSDEs)的理论基础;之后,正倒向随机微分方程组得到了广泛研究,并被应用于众多研究领域中,如随机最优控制、偏微分方程、金融数学、风险度量、非线性期望等.近年来,正倒向随机微分方程组的数值求解研究获得了越来越多的关注,本文旨在基于正倒向随机微分方程组的特性,介绍正倒向随机微分方程组的主要数值求解方法.我们将重点介绍讨论求解FBSDEs的积分离散法和微分近似法,包括一步法和多步法,以及相应的数值分析和理论分析结果.微分近似法能构造出求解全耦合FBSDEs的高效高精度并行数值方法,并且该方法采用最简单的Euler方法求解正向随机微分方程,极大地简化了问题求解的复杂度.文章最后,我们尝试提出关于FBSDEs数值求解研究面临的一些亟待解决和具有挑战性的问题.     

A Family of Fifth-Order Iterative Methods for Solving Nonlinear Equations

In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order five. Numerical examples show that the new methods are comparable with the well known existing methods and give better results in many aspects.     

Discontinuous Galerkin Methods for Solving Two Membranes Problem

The two membranes problem is known as a free boundary problem, which arises from a variety of applications. In this article we extend the ideas in the article [Wang, Han, and Cheng, SIAM J. Numer. Anal. 48 (2010), 703–733] to use discontinuous Galerkin methods to solve the two membranes problem. A priori error estimates are established, which reach optimal convergence order for linear elements.     

Laguerre Tau Methods for Solving Higher-Order Ordinary Differential Equations

In this paper, we present two numerical methods for solving higher-order differential equations using the Laguerre Tau method. These methods generate linear systems, which can be solved by Gauss elimination with maximal partial pivoting strategy. Results of some numerical experiments and theoretical analysis are presented.     

Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels

The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation.