The Maximum Principle for Time-Fractional Diffusion Equations and Its Application

Using an equivalent reformulation of the definition of the Caputo fractional derivative, we present the stability and convergence analysis for a finite difference scheme that we use to solve a time-fractional diffusion equation. The analysis is based on a maximum principle for such equations.     

On preserving long-time features of a linear stochastic oscillator

A method for the numerical solution of stochastic differential equations is presented. The method has mean-square order equal to 1/2 when it is applied to a general stochastic differential equation and equal to 1 if the equation has additive noise. In addition, it is shown that the method captures some long-time properties of a linear stochastic oscillator: It reproduces exactly the growth rate of the second moment and the oscillation property of the solution. AMS subject classification (2000)  60H10, 34F05, 65U05, 60K40     

Complete weight enumerators of a class of linear codes with two weights

In this paper, a class of $p$-ary linear codes with two weights is constructed by using the properties of cyclotomic classes of ${\mathbb{F}}_{p^2}^1$. The complete weight enumerators of these linear codes are also determined. In some cases, they are optimal and can be employed to obtain secret sharing schemes with interesting access structures and asymptotically optimal systematic authentication codes.     

Integrated production and delivery scheduling with disjoint windows

Consider a company that manufactures perishable goods. The company relies on a third party to deliver goods, which picks up delivery products at regular or irregular times. At each delivery time, there is a time window that products can be produced to be delivered at that delivery time. The time windows are disjoint. Suppose we have a set of jobs with each job specifying its delivery time, processing time and profit. The company can earn the profit if the job is produced and delivered at its specified delivery time. From the company point of view, we are interested in picking a subset of jobs to produce and deliver so as to maximize the total profit. The unpicked jobs will be discarded without penalty. We consider both the single machine case and the parallel and identical machine case.In this article we consider three kinds of profits: (1) arbitrary profit, (2) equal profit, and (3) profit proportional to its processing time. In the first case, we give a fully polynomial time approximation scheme (FPTAS) for a single machine with running time . Using the bound improvement technique of Kovalyov, the running time can be further reduced to . In the second case, we give an O(nlogn)-time optimal algorithm for a single machine. In the third case, we give an FPTAS for a single machine with running time . All of our algorithms can be extended to parallel and identical machines with a degradation of performance ratios.     

带弱奇异核非线性积分微分方程的有限元分析

讨论了带弱奇异核的非线性抛物积分微分方程的Hermite型各向异性矩形元逼近.在各向异性网格下导出了关于Riesz投影的L~2和H~1模的误差估计.在半离散和向后欧拉全离散格式下,基于Riesz投影的性质并利用平均值技巧,分别得到了L~2模意义下的最优误差估计.     

Explicit finite difference schemes with extended stability for advection equations

In this study an explicit central difference approximation of the generalized leap-frog type is applied to the one- and two-dimensional advection equations. The stability of the considered numerical schemes is investigated and the scheme with the largest stable time step is found. For the linear and nonlinear advection equations numerical experiments with different schemes from the considered class are performed in order to evaluate the practical stability of the designed schemes.     

部分权重信息下的区间数多指标灰靶决策模型

针对决策信息为区间数且部分权重信息已知的多指标决策问题,对现有的灰靶决策模型进行了拓展研究。首先建立了区间数灰靶和靶心距,并基于所有方案综合属性值最大化的准则,对各方案进行局部优化,然后采用两阶段法求出最佳协调向量,根据归一化的组合权向量计算各方案的综合属性值,并据此进行排序与决策。最后以实例验证了新模型的有效性与实用性。     

Exact finite-difference methods based on nonlinear means

The necessary and sufficient condition for an autonomous, ordinary differential equation to be exactly solved by a given Evans-Sanugi, nonlinear, one-step, finite difference method based on ‘classical means’ are derived. A necessary, but not sufficient, condition yields the most general nonlinearity for the differential equation which is independent of the step size. Examples of differential equations for which either nonlinear trapezoidal or nonlinear implicit midpoint methods based on arithmetic, harmonic, contraharmonic, quadratic, geometric, Heronian, centroidal and logarithmic means are exact, are presented. These new exact difference schemes may be useful in future developments of new Denk-Bulirsch, Le-Roux or Kojouhavov-Chen schemes for nonlinear evolution equations with or without blow-up.