On the Mean Square Formula of the Error Termfor a Class of Arithmetical Functions

 Based on the method in Meurman [5], we study the mean square formula of the error term for a class of Arithmetical functions whose Dirichlet series satisfies a functional equation with multiple gamma factors. We obtain improvements on some results of Chandrasekharan and Narasimhan [1].     

灰色系统模型的优化岭回归算法

文献[1]指出了目前用普通最小二乘法估计灰微分方程参数的方法由于方程组的病态问题很难求解得合理的参数;文献[2]指出了根据初值求解灰色系统模型的时间响应式的方法由于初值的误差使所求得时间响应式产生系统误差.为了克服灰色模型的上述两个缺点,本文设计了一种求解灰色系统模型的优化岭回归算法,计算一个广泛引用的算例演示了这种算法的优越性.     

Improvement in estimating the population mean in simple random sampling using information on auxiliary attribute

This paper proposes some estimators for the population mean by adapting the estimator in Singh et al. (2008) [5] to the ratio estimators presented in Kadilar and Cingi 2006 [2]. We obtain mean square error (MSE) equation for all proposed estimators, and show that all proposed estimators are always more efficient than ratio estimator in Naik and Gupta (1996) [3], and Singh et al. (2008) [5]. The results have been illustrated numerically by taking some empirical population considered in the literature.     

On the Mean Square Formula of the Error Termfor a Class of Arithmetical Functions

 Based on the method in Meurman [5], we study the mean square formula of the error term for a class of Arithmetical functions whose Dirichlet series satisfies a functional equation with multiple gamma factors. We obtain improvements on some results of Chandrasekharan and Narasimhan [1]. (Received 29 June 1998; in final form 11 November 1998)     

多孔直杆扭转问题的边界元分析

多孔直杆弹性扭转问题导致一个Poisson方程的非局部边值问题,相应的积分方程既包含未知函数又包含若干个未知数,并带有约束条件,我们通过把问题化为若干个无约束的变分问题证明了变分解的唯一性,并对近似解进行了误差分析,得到了最佳的误差估计。     

High-order filtered scheme for front propagation problems

In this work we develop a specific application of the scheme proposed and analyzed in [1] to front propagation problems. The approach is based on the level-set method which leads in the isotropic case to a classical evolutive first order Hamilton- Jacobi equation.We will apply to this equation high-order “filtered schemes”, for these schemes the strong monotonicity property will not be satisfied. However, a weak ?-monotonicity property applies and this is enough to obtain a convergence result and a precise error estimate. In the last section we will present several examples where we solve front propagation problems by filtered scheme in two and three dimensions showing the accuracy of our method.     

Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.     

Optimal homotopy analysis and control of error for solutions to the non-local Whitham equation

The Whitham equation is a non-local model for nonlinear dispersive water waves. Since this equation is both nonlinear and non-local, exact or analytical solutions are rare except for in a few special cases. As such, an analytical method which results in minimal error is highly desirable for general forms of the Whitham equation. We obtain approximate analytical solutions to the non-local Whitham equation for general initial data by way of the optimal homotopy analysis method, through the use of a partial differential auxiliary linear operator. A method to control the residual error of these approximate solutions, through the use of the embedded convergence control parameter, is discussed in the context of optimal homotopy analysis. We obtain residual error minimizing solutions, using relatively few terms in the solution series, in the case of several different kernels and associated initial data. Interestingly, we find that for a specific class of initial data, there exists an exact solution given by the first term in the homotopy expansion. A specific example of initial data which satisfies the condition producing an exact solution is included. These results demonstrate the applicability of optimal homotopy analysis to equations which are simultaneously nonlinear and non-local.     

A new equation on the Calabi-Yau metrics in low dimensions

We obtain an equation on the metrics of compact Kähler manifolds in dimensions 2 and 3, whose solutions are Calabi-Yau metrics. This equation differs from the Monge-Ampère equation considered by Calabi [1] and Yau [2].     

不可压缩核废料污染问题的变网格特征有限元法

1 引言 多孔介质中的核废料污染问题是环境保护领域的重要课题。对于不可压缩二维模型,它是地层中迁移型耦合抛物型方程组的初边值问题:     

Finite element scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient

A finite element scheme is considered for the viscous Cahn-Hilliard equation with the nonconstant gradient energy coefficient. The scheme inherits energy decay property and mass conservation as for the classical solution. We obtain the corresponding error estimate using the extended Lax-Richtmyer equivalence theorem.     

On the convergence of the method for indefinite integration of oscillatory and singular functions

We consider the problem of convergence and error estimation of the method for computing indefinite integrals proposed in Keller [8]. To this end, we have analysed the properties of the difference operator related to the difference equation for the Chebyshev coefficients of a function that satisfies a given linear differential equation with polynomial coefficients. Properties of this operator were never investigated before. The obtained results lead us to the conclusion that the studied method is always convergent. We also give a rigorous proof of the error estimates.     

Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the so-called aggregation equation, by employing 2-Wasserstein distance. By introducing an intermediate system, given by the pressureless Euler equations with nonlocal forces, we can quantify the error between the spatial densities of the kinetic equation and the pressureless Euler system by means of relative entropy type arguments combined with the 2-Wasserstein distance. This together with the quantitative error estimate between the pressureless Euler system and the aggregation equation in 2-Wasserstein distance in [Commun. Math. Phys, 365, (2019), 329–361] establishes the quantitative bounds on the error between the kinetic equation and the aggregation equation.     

Economical Iterative Algorithms for Solving Stationary Problems of Mathematical Physics

We construct and investigate additive iterative methods of complete approximation for solving stationary problems of mathematical physics. We prove the convergence of the proposed methods and obtain error estimates without the requirement of commutativity of the decomposition operators. We provide the results of a computational experiment for a three-dimensional boundary-value problem. We consider possible generalizations of algorithms for equations with mixed derivatives and Navier–Stokes equation systems.     

Pointwise error estimates for a generalized Oseen problem and an application to an optimal control problem

We derive pointwise error estimates for a generalized Oseen when it is approximated by a low order Taylor‐Hood finite element scheme in two dimensions. The analysis is based on estimates for regularized Green's functions associated with a generalized Oseen problem on weighted Sobolev spaces and weighted interpolation results. We apply the maximum norm results to obtain convergence in an optimal control problem governed by a generalized Oseen equation and present a numerical example that allows us to show the behavior of the error.     

Error estimates for stochastic differential games: the adverse stopping case

^** Email: frederic.bonnans{at}inria.fr^*** Email: stefania.maroso{at}inria.fr^**** Email: zidani{at}ensta.fr We obtain error bounds for monotone approximation schemes ofa particular Isaacs equation. This is an extension of the theoryfor estimating errors for the Hamilton–Jacobi–Bellmanequation. To obtain the upper error bound, we consider the ‘Krylovregularization’ of the Isaacs equation to build an approximatesub-solution of the scheme. To get the lower error bound, weextend the method of Barles & Jakobsen (2005, SIAM J. Numer.Anal.) which consists in introducing a switching system whosesolutions are local super-solutions of the Isaacs equation.     

局部加密的非标准混杂五点矩形元分析

In this paper we apply the nonconforming theory in Z.C.Shi to construct a hybrid 5-node finite element satisfying the F1-patch test. We show that the hybrid 5-node finitc element for two-order elliptic equation is convergent and obtain an error estimate.     

五次NLS方程全离散谱格式的大时间性态

Nonlinear Schroedinger equation arises in many physical problems. There are many works in which properties of the solution are studied. In this paper we use fully discrete Fourier spectral method to get an approximation solution of nonlinear weakly dissipative Schroedinger equation with quintic term. We give a large-time error estimate and obtain the existence of the approximate attractor A N^k.     

第一类不适定算子方程的解的表示

该文在L2中讨论了第一类算子方程Au＝f当A-1无定义和A-1不是单值的情形下的不适定求解问题，给出了解存在的充要条件，当有解时，得到了形式解，多解时形式解就是最小范数解，并且得到了近似解表达式，给出了误差估计．     

Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates

In this paper,we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates.We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation.We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions.Furthermore,we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation.We obtain two critical values of r,and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value,while it appears as a damped oscillatory wave if |r| is less than some critical value.By means of analysis and the undetermined coefficients method,we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation.Based on the above discussions and according to the evolution relations of orbits in the global phase portraits,we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method.Finally,using the homogenization principle,we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions.Moreover,we also give the error estimates for these approximate solutions.