Convergence ball and error analysis of Ostrowski-Traub’s method

Under the hypotheses that the second-order and third-order derivatives of a function are bounded, an estimate of the radius of the convergence ball of Ostrowski-Traub＇s method is obtained. An error analysis is given which matches the convergence order of the method. Finally, two examples are provided to show applications of our theorem.

A Generalized Lyapunov-Sylvester Computational Method for Numerical Solutions of NLS Equation with Singular Potential

In the present paper a numerical method is developed to approximate the solution of two-dimensional Nonlinear Schrodinger equation in the presence of a sin- gular potential. The method leads to generalized Lyapunov-Sylvester algebraic opera- tors that are shown to be invertible using original topological and differential calculus issued methods. The numerical scheme is proved to be consistent, convergent and sta- ble using the Lyapunov criterion, lax equivalence theorem and the properties of the generalized Lyapunov-Sylvester operators.

Attractive Multistep Reproducing Kernel Approach for Solving Stiffness Differential Systems of Ordinary Differential Equations and Some Error Analysis

In this paper,an efficient multi-step scheme is presented based on reproducing kernel Hilbert space(RKHS)theory for solving ordinary stiff differential systems.The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform formfor a rapidly convergent series in the posed Sobolev space.Using the Gram-Schmidt orthogonality process,complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction.Consequently,by applying the standard RKHS method to each subinterval,approximate solutions that converge uniformly to the exact solutions are obtained.For this purpose,several numerical examples are tested to show proposed algorithm’s superiority,simplicity,and efficiency.The gained results indicate that themulti-step RKHSmethod is suitable for solving linear and nonlinear stiffness systems over an extensive duration and giving highly accurate outcomes.

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term

This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.

Influence of dissipation on solitary wave solution to generalized Boussinesq equation

This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves.The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail.The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied,and the critical values that can describe the magnitude of the dissipation effect are,respectively,found for the two cases of b_3<0 and b_3>0 in the equation.The results show that,when the dissipation effect is significant(i.e.,r is greater than the critical value in a certain situation),the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution;when the dissipation effect is small(i.e.,r is smaller than the critical value in a certain situation),the traveling wave solution to the equation appears as the oscillation attenuation solution.By using the hypothesis undetermined method,all possible solitary wave solutions to the equation when there is no dissipation effect(i.e.,r=0)and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained;in particular,when the dissipation effect is small,an approximate solution of the oscillation attenuation solution can be achieved.This paper is further based on the idea of the homogenization principles.By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper,and by investigating the asymptotic behavior of the solution at infinity,the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained,which is an infinitesimal amount that decays exponentially.The influence of the dissipation coefficient on the amplitude,frequency,period,and energy of the bounded traveling wave solution of the equation is also discussed.

Abel积分方程的数值解

文章研究了Abel积分方程的数值解。首先对Abel积分方程进行多次积分得到与之等价的积分方程,结合分段泰勒级数得到方程的数值解;然后利用Banach空间的映射原理,证明数值解的收敛性;最后通过例题验证此方法的可行性与有效性,并给相应的出误差估计。

Numerical Study for a Class of Variable Order Fractional Integral-differential Equation in Terms of Bernstein Polynomials

The aim of this paper is to seek the numerical solution of a class of variable order fractional integral-differential equation in terms of Bernstein polynomials.The fractional derivative is described in the Caputo sense.Four kinds of operational matrixes of Bernstein polynomials are introduced and are utilized to reduce the initial equation to the solution of algebraic equations after dispersing the variable.By solving the algebraic equations,the numerical solutions are acquired.The method in general is easy to implement and yields good results.Numerical examples are provided to demonstrate the validity and applicability of the method.

BOUNDED TRAVELING WAVE SOLUTIONS OF VARIANT BOUSSINESQ EQUATION WITH A DISSIPATION TERM AND DISSIPATION EFFECT

This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and general shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r^＊ which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if ｜r｜≥ r^＊; while they appear as damped oscillatory waves if ｜r｜ 〈 r^＊. We also obtain kink profile solitary wave solutions with and without dissipation effect. On the basis of the above discussion, we sensibly design the structure of the approximate damped oscillatory solutions according to the orbits evolution relation corresponding to the component u（ξ） in the global phase portraits, and then obtain the approximate solutions （u（ξ）, H（ξ））. Furthermore, by using homogenization principle, we give their error estimates by establishing the integral equation which reflects the relation between exact and approximate solutions. Finally, we discuss the dissipation effect on the amplitude, frequency, and energy decay of the bounded traveling wave solutions.

Three-Variable Shifted Jacobi Polynomials Approach for Numerically Solving Three-Dimensional Multi-Term Fractional-Order PDEs with Variable Coefficients

In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.

Finite Element Analysis of the Ramberg-Osgood Bar

In this work, we present a priori error estimates of finite element approximations of the solution for the equilibrium equation of an axially loaded Ramberg-Osgood bar. The existence and uniqueness of the solution to the associated nonlinear two point boundary value problem is established and used as a foundation for the finite element analysis.

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.

On the problem of the dynamical reactions of a rolling wheelset to real track irregularities

We investigate numerically the dynamical reactions of a moving wheelset model to real measured track irregularities.The background is to examine whether the dynamics are suitable as the input to the inverse problem:determine the true track geometry from measured wheelset dynamical reactions.It is known that the method works well for the vertical position of the rails but the computed lateral position is often flawed.We find that the lateral motion of the wheelset often may differ from the track geometry.The cases are investigated closely but the reasons remain unknown.While the wheelset dynamics reflect the larger(>4-6 mm)aperiodic track disturbances and single large disturbances quite well,this does not seem to be the case for general smaller or periodic track irregularities or sections behind single large disturbances.The resulting dynamics of a wheelset to lateral track irregularities are in general not sufficiently accurate to be used as the basis for a description of the track irregularities.

基于神经网络和演化算法的土石坝位移反演分析

构造了基于综合应用人工神经网络和演化算法的位移反演分析方法。该法使用具有较强非线性映射能力的神经网络模型代替有限元计算,提高了计算效率。采用演化算法和Vogl快速算法,同时优化神经网络的结构和权值,增加其适应性并加快训练速度;使用多种群演化等策略,改善演化算法的全局收敛性和收敛速度。以三峡茅坪溪防护土石坝的变形反演分析为例,研究了神经网络演化代数以及训练样本数量对神经网络模拟能力的影响,证明了所建立的反演分析方法的有效性。

Adomian分解法求解非线性分数阶Fredholm积分微分方程

为了求解一类非线性分数阶Fredholm积分微分方程的数值解,本文将Adomian分解法(Adomian Decomposition Method,ADM)引入到非线性分数阶Fredholm微积分方程的求解中.将ADM多项式与分数阶积分定义有效结合,得到Adomian级数解.通过收敛性分析证明所得的级数解收敛于精确解,给出最大绝对截断误差.并结合实例,证明方法的有效性和实用性.

一种基于模型误差预测的UKF方法

UnscentedKalman滤波器(UKF)对本质非线性系统具有估计精度高、收敛速度快和容易实现等优点,但是对系统的模型误差比较敏感。针对这一问题,提出了一种基于模型误差预测的UKF方法,称为PUKF(PredictiveUnscentedKalmanFilter)。它利用非线性预测滤波器(NPF)的模型误差预测过程,能够对不准确的系统模型进行实时修正,弥补了UKF方法的不足。仿真结果表明,相对于原始的UKF方法,新方法从滤波精度、收敛速度和收敛的稳定性等几个方面,显著提高了非线性滤波的性能。PUKF可适用于模型不确定、非线性较强系统的滤波。

Adomian分解法求解非线性分数阶积分微分方程

求一类非线性分数阶Volterra积分微分方程数值解,给出了Adomian分解法.将Adomian多项式与分数阶积分定义有效结合,得到了Adomian级数解.收敛性分析证明了所得级数解收敛于精确解,并给出最大截断误差.结果表明:随着Adomian多项式个数的增加,数值解的精度也越来越高.数值算例表明了该方法的可行性和有效性.与已有的方法相比,Adomian分解法操作更有效、更方便.

带扩散项色散方程的交替分组差分方法(英文)

给出了逼近带扩散项色散方程的非对称差分格式,并用这些差分公式构造了求解色散方程的交替分段差分算法,算法是无条件稳定的,能直接在并行机上使用,并给出了模型问题的试验结果.

转角信息未知条件下的结构参数识别方法研究

针对结构中的梁式构件提出了一种基于反应曲线拟合的参数识别方法。该方法采用一组正交基函数对Euler-Bernoulli梁在不同时刻的平动反应曲线进行拟合,继而通过平动反应曲线对测点坐标的导数获得梁的转角反应曲线。为解决反应曲线拟合过程中的噪声淹没现象,提出了一种样本平均方法以消除噪声的影响,该方法利用采样集合的子集的平均形成的序列进行识别。算例分析表明,反应曲线拟合方法可较为准确地获得梁式构件的转角反应,且样本平均方法可大大降低参数识别过程中噪声的影响。

次级通道模型误差下滤波X型最小均方差算法收敛性分析

给出了滤波Ｘ型ＬＭＳ（ＦＸＬＭＳ）算法收敛的一个充分条件，指出如果次级通道传递函数与其估计值在所有频带上满足正实条件，则ＦＸＬＭＳ算法对任意参考信号收敛。若上述正实条件仅在某些频带上满足，则ＦＸＬＭＳ算法的收敛将依赖于参考信号功率谱密度的分布。收敛步长取决于某特定相关矩阵特征值的分布。将上述结论应用于时延型ＬＭＳ（ＤＬＭＳ）算法，得出在时延估计存在误差时，ＤＬＭＳ算法收敛于若干离散的频带，而频带宽度完全取决于“对延估计误差频率”（时延估计误差倒数的１／４）。

分区混合元法分析平面裂纹问题

基于分区混合能量原理的分区混合元法是一种高精度有限元方法,可用于分析含裂纹、孔洞、切口等缺陷的问题。本文推导了分区混合元法中应力元的刚度矩阵;分析了应力元的多余零能模式,并证明了整体分析中对多余零能模式的消弭;文中还对分区混合元法中应力元的应力项数与应力单元尺寸对分析结果的影响进行了系统的讨论。