Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method

This paper investigates the numerical solutions of singular second order three-point boundary value problems using reproducing kernel Hilbert space method. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel Hilbert space method cannot be used directly to solve a singular second order three-point boundary value problem, so we convert it into an equivalent integro-differential equation, which can be solved using reproducing kernel Hilbert space method. Four numerical examples are given to demonstrate the efficiency of the present method. The numerical results demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

Oscillation of even order nonlinear delay dynamic equations on time scales

One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor’s Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is sufficient for oscillation of even order dynamic equations on time scales. The arguments are based on Taylor monomials on time scales.     

An Initial Value Technique for Singularly Perturbed Convection–Diffusion Problems with a Negative Shift

In this paper, a numerical method named as Initial Value Technique (IVT) is suggested to solve the singularly perturbed boundary value problem for the second order ordinary differential equations of convection–diffusion type with a delay (negative shift). In this technique, the original problem of solving the second order equation is reduced to solving two first order differential equations, one of which is singularly perturbed without delay and other one is regular with a delay term. The singularly perturbed problem is solved by the second order hybrid finite difference scheme, whereas the delay problem is solved by the fourth order Runge–Kutta method with Hermite interpolation. An error estimate is derived by using the supremum norm. Numerical results are provided to illustrate the theoretical results.     

二阶微分方程第二边值问题模糊状态的数值求解

本讨论二阶微分方程的第二边值问题具有模糊不确定性时，运用模糊仿真原理和差分方法，求其边值问题的数值解法。     

An unconstrained optimization method using nonmonotone second order Goldstein’s line search

In this paper, an unconstrained optimization method using the nonmonotone second order Goldstein’s line search is proposed. By using the negative curvature information from the Hessian, the sequence generated is shown to converge to a stationary point with the second order optimality conditions. Numerical tests on a set of standard test problems confirm the efficiency of our new method. This work was supported by the National Natural Science Foundation of China (Grant No. 10231060) and the Specialized Research Fund of Doctoral Program of Higher Education of China (Grant No. 20040319003)     

PRECONDITIONING MIXED ELEMENT METHOD FOR NONSELFADJOINT AND INDEFINITE SECOND ORDER ELLIPTIC PROBLEMS*

In this paper, an equivalence between mixed element method and nonconforming element method for nonselfad joint and indefinite second order elliptic problems is established without using any bubble functions. It is proved that the H~1-condition number of preconditioned operator B_h~(-1)A_h is uniformly bounded and its B_h-singular values cluster in a positive finite interval, where A_h is the equivalent nonconforming element discretization of nonselfad joint and indefinite second order elliptic operator A, B_h is usual noncon forming element discretization of selfadjoint and positive definite second order elliptic operator B. Finally a simple V-cycle multigrid implementation of B_h~(-1) is given.     

A reduced Hessian algorithm with line search filter method for nonlinear programming

This paper proposes a line search filter reduced Hessian method for nonlinear equality constrained optimization. The feature of the presented algorithm is that the reduced Hessian method is used to produce a search direction, a backtracking line search procedure to generate step size, some filtered rules to determine step acceptance, second order correction technique to reduce infeasibility and overcome the Maratos effects. It is shown that this algorithm does not suffer from the Maratos effects by using second order correction step, and under mild assumptions fast convergence to second order sufficient local solutions is achieved. The numerical experiment is reported to show the effectiveness of the proposed algorithm.     

Convergence of compact ADI method for solving linear Schrödinger equations

A compact ADI scheme of second‐order in time and fourth‐order in space is proposed for solving linear Schrödinger equations with periodic boundary conditions. By using the recently suggested discrete energy method, it is shown that the stable compact ADI method is unconditionally convergent in the maximum norm. Numerical experiments, including the comparisons with the second‐order ADI scheme and the time‐splitting Fourier pseudospectral method, are presented to support the theoretical results and show the effectiveness of our method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Application of reproducing kernel method to third order three-point boundary value problems

This paper investigates the analytical approximate solutions of third order three-point boundary value problems using reproducing kernel method. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve third order three-point boundary value problems, since there is no method of obtaining reproducing kernel satisfying three-point boundary conditions. This paper presents a method for solving reproducing kernel satisfying three-point boundary conditions so that reproducing kernel method can be used to solve third order three-point boundary value problems. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

关于二层海中的二阶波浪绕射问题*

在本文中,我们用二层海模型,探讨了层化海洋中任意三维物体的二阶波浪绕射问题,给出了多色波场中二阶波浪散射势边值问题的数学提法以及基于一个弱的远场辐射条件下解的表式。同时,利用Green定理,并通过引入一个辅助势函数,我们导出了结构所受二阶波浪荷载的积分表式。结果表明,海水的层化特性对结构物所受之二阶差频波浪荷载可能具有显著的影响。     

On convergence of a new secant-like method for solving nonlinear equations

In this paper, we prove that the order of a new secant-like method presented recently with the so-called order of 2.618 is only 2.414. Some mistakes in the derivation leading to such a conclusion are pointed out. Meanwhile, under the assumption that the second derivative of the involved function is bounded, the convergence radius of the secant-like method is given, and error estimates matching its convergence order are also provided by using a generalized Fibonacci sequence.     

Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problems

In this letter, a new numerical method is proposed for solving second order linear singularly perturbed boundary value problems with left layers. Firstly a piecewise reproducing kernel method is proposed for second order linear singularly perturbed initial value problems. By combining the method and the shooting method, an effective numerical method is then proposed for solving second order linear singularly perturbed boundary value problems. Two numerical examples are used to show the effectiveness of the present method.     

Vibration control of micro-scale structures using their reduced second order bilinear models based on multi-moment matching criteria

This paper deals with vibration control of micro-scale structures; i.e. MEMS devices. For modeling of the structures, finite element method which is a distinguished and accurate technique will be used. This method, however, leads to a model with high number of degrees of freedom which may cause computational costs especially for control problems. Hence, we will apply the second order Krylov subspace method based on multi-moment matching to obtain a reduced order model which is in the form of a second order bilinear system. For vibration suppression of the corresponding micro-structure, a quadratic feedback controller and also a linear state feedback controller using linear matrix inequality (LMI) will be designed. Finally, a micro-cantilever beam will be considered as a practical case study and simulation results of applying the proposed method will be presented.     

Banach空间中两点边值问题解的存在唯一性

利用半序方法，得到Ｂａｎａｃｈ空间中二阶常微分方程两点边值问题解的存在唯一性定理．     

A secant algorithm with line search filter method for nonlinear optimization

Filter methods were initially designed for nonlinear programming problems by Fletcher and Leyffer. In this paper we propose a secant algorithm with line search filter method for nonlinear equality constrained optimization. The algorithm yields the global convergence under some reasonable conditions. By using the Lagrangian function value in the filter we establish that the proposed algorithm can overcome the Maratos effect without using second order correction step, so that fast local superlinear convergence to second order sufficient local solution is achieved. The primary numerical results are presented to confirm the robustness and efficiency of our approach.     

Double asymptotic expansion in the problem of radiation heat exchange of annular fins of trapezoidal shape

We propose a method of double asymptotic expansion of an approximate analytic solution of the problem on thermal irradiation of annular radiator fins with a trapezoidal cross-section, which is reduced to a nonlinear differential equation of the second order with variable coefficients. The method is based on using the Poincaré small parameter method and method of phase integrals.     

多体系统动力学微分／代数方程组的一类新的数值分析方法

本文讨论了多体系统动力学微分／代数混合方程组的数值离散问题．首先把参数t并入广义坐标讨论，简化了方程组及其隐含条件的结构，并将其化为指标1的方程组．然后利用方程组的特殊结构，引入一种局部离散技巧并构造了相应的算法．算法结构紧凑，易于编程，具有较高的计算效率和良好的数值性态，且其形式适合于各种数值积分方法的的实施．文末给出了具体算例．     

A second‐order partitioned method with different subdomain time steps for the evolutionary Stokes‐Darcy system

A second‐order decoupled algorithm for the nonstationary Stokes‐Darcy system, which allows different time steps in two subregions, is proposed and analyzed in this paper. The algorithm, which is a combination of the second‐order backward differentiation formula and second‐order extrapolation method, uncouples the problem into two decoupled problems per time step. We prove the unconditional stability and long‐time stability of the decoupled scheme with different time steps and derive error estimates of this decoupled algorithm using finite element spatial discretization. Numerical experiments are provided to illustrate the accuracy, effectiveness, and stability of the decoupled algorithm and show its advantages of increasing accuracy and efficiency.     

A predictor-corrector scheme for the numerical solution of the Boussinesq equation

A fourth order in time and second order in space scheme using a finite-difference method is developed for the non-linear Boussinesq equation. For the solution of the resulting non-liner system a predictor-corrector pair is proposed. The method is analyzed for local truncation error and stability. The results of a number of numerical experiments for both the single and the double-soliton waves are given.     

Macroscopic modeling and simulations of room evacuation

We analyze numerically two macroscopic models of crowd dynamics: the classical Hughes model and the second order model being an extension to pedestrian motion of the Payne–Whitham vehicular traffic model. The desired direction of motion is determined by solving an eikonal equation with density dependent running cost, which results in minimization of the travel time and avoidance of congested areas. We apply a mixed finite volume-finite element method to solve the problems and present error analysis for the eikonal solver, gradient computation and the second order model yielding a first order convergence. We show that Hughes’ model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit.