A random Euler scheme for Carathéodory differential equations

We study a random Euler scheme for the approximation of Carathéodory differential equations and give a precise error analysis. In particular, we show that under weak assumptions, this approximation scheme obtains the same rate of convergence as the classical Monte–Carlo method for integration problems.     

A simple proof for persistence of snap-back repellers

In this article, we show that if f has a snap-back repeller then any small C¹ perturbation of f has a snap-back repeller, and hence has Li-Yorke chaos and positive topological entropy, by simply using the implicit function theorem. We also give some examples.     

Direct numerical method for an inverse problem of a parabolic partial differential equation

A coefficient inverse problem of the one-dimensional parabolic equation is solved by a high-order compact finite difference method in this paper. The problem of recovering a time-dependent coefficient in a parabolic partial differential equation has attracted considerable attention recently. While many theoretical results regarding the existence and uniqueness of the solution are obtained, the development of efficient and accurate numerical methods is still far from satisfactory. In this paper a fourth-order efficient numerical method is proposed to calculate the function u(x,t) and the unknown coefficient a(t) in a parabolic partial differential equation. Several numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.     

Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings

In this paper, we consider the weak and strong convergence of an implicit iterative process with errors for two finite families of asymptotically nonexpansive mappings in the framework of Banach spaces. Our results presented in this paper improve and extend the recent ones announced by many others.     

A PARALLEL COMPUTATION SCHEME FOR IMPLICIT RUNGE-KUTTA METHODS AND THE ITERATIVELY B-CONVERGENCE OF ITS NEWTON ITERATIVE PROCESS

In this paper, based on the implicit Runge-Kutta(IRK) methods, we derive a class of parallel scheme that can be implemented on the parallel computers with Ns(N is a positive even number) processors efficiently, and discuss the iteratively B-convergence of the Newton iterative process for solving the algebraic equations of the scheme, secondly we present a strategy providing initial values parallelly for the iterative process. Finally, some numerical results show that our parallel scheme is higher efficient as N is not so large.     

A two-level finite element method for the Navier–Stokes equations based on a new projection

We consider a fully discrete two-level approximation for the time-dependent Navier–Stokes equations in two dimension based on a time-dependent projection. By defining this new projection, the iteration between large and small eddy components can be reflected by its associated space splitting. Hence, we can get a weakly coupled system of large and small eddy components. This two-level method applies the finite element method in space and Crank–Nicolson scheme in time. Moreover,the analysis and some numerical examples are shown that the proposed two-level scheme can reach the same accuracy as the classical one-level Crank–Nicolson method with a very fine mesh size h by choosing a proper coarse mesh size H. However, the two-level method will involve much less work.     

An approximation-based approach for fuzzy multi-period production planning problem with credibility objective

This paper develops a fuzzy multi-period production planning and sourcing problem with credibility objective, in which a manufacturer has a number of plants or subcontractors. According to the credibility service levels set by customers in advance, the manufacturer has to satisfy different product demands. In the proposed production problem, production cost, inventory cost and product demands are uncertain and characterized by fuzzy variables. The problem is to determine when and how many products are manufactured so as to maximize the credibility of the fuzzy costs not exceeding a given allowable invested capital, and this credibility can be regarded as the investment risk criteria in fuzzy decision systems. In the case when the fuzzy parameters are mutually independent gamma distributions, we can turn the service level constraints into their equivalent deterministic forms. However, in this situation the exact analytical expression for the credibility objective is unavailable, thus conventional optimization algorithms cannot be used to solve our production planning problems. To overcome this obstacle, we adopt an approximation scheme to compute the credibility objective, and deal with the convergence about the computational method. Furthermore, we develop two heuristic solution methods. The first is a combination of the approximation method and a particle swarm optimization (PSO) algorithm, and the second is a hybrid algorithm by integrating the approximation method, a neural network (NN), and the PSO algorithm. Finally, we consider one 6-product source, 6-period production planning problem, and compare the effectiveness of two algorithms via numerical experiments.     

A new branch and bound algorithm for cell formation problem

Cell formation (CF) is the first and the most important problem in designing cellular manufacturing systems. Due to its non-polynomial nature, various heuristic and metaheuristic algorithms have been proposed to solve CF problem. Despite the popularity of heuristic algorithms, few studies have attempted to develop exact algorithms, such as branch and bound (B&B) algorithms, for this problem. We develop three types of branch and bound algorithms to deal with the cell formation problem. The first algorithm uses a binary branching scheme based on the definitions provided for the decision variables. Unlike the first algorithm, which relies on the mathematical model, the second one is designed based on the structure of the cell formation problem. The last algorithm has a similar structure to the second one, except that it has the ability to eliminate duplicated nodes in branching trees. The proposed branch and bound algorithms and a hybrid genetic algorithm are compared through some numerical examples. The results demonstrate the effectiveness of the modified problem-oriented branch and bound algorithm in solving relatively large size cell formation problems.     

一类非线性Schr dinger方程的高精度守恒数值格式

1引言本文讨论下面非线性Schr(?)dinger方程(NLS)方程的初边值问题:i(?)u/(?)t (?)~2u/(?)x~2 2|u~2|u=0,(1) u(x_l,t)=u(x_r,t)=0,t>0,(2) u(x,0)=u_0(x),x_l≤x≤x_r,(3)其中u(x,t)是复值函数,u_0(x)为已知的复值函数,i~2=-1.该问题有着如下的电荷与能量守恒关系: