脉冲微分混沌系统动力学数值模拟与分析

由于脉冲微分混沌系统具有复杂的性态,在理论分析时具有一定的难度,而数值分析在一定程度上可以提供一些指导,所以数值模拟方法成为脉冲微分混沌系统研究的重要手段.该文设计了脉冲微分混沌系统的动力学分析算法,并将数值解以可视化的形式展现,绘制出方程组解的相图、分岔图、Poincaré截面.以具有Holling type-II功...     

一类Leslie-Gower捕食-食饵模型的定性分析

研究一类具有Holling-II型反应函数的Leslie-Gower捕食-食饵模型。给出了平衡态方程解的先验估计,讨论了正常数解的局部渐近稳定性和全局渐近稳定性,利用分歧理论,得到了局部分歧解的存在性,最后将局部分歧延拓为全局分歧。     

非线性延迟积分微分方程线性多步法的渐近稳定性

本文研究了非线性延迟积分微分方程线性多步法的渐近稳定性.证明了在约束网格下,带有复合求积公式A-稳定的线性多步法能够保持解析解的渐近稳定性.文章最后,数值试验验证了本文的结论.     

H∞滤波问题数值求解的精细积分算法

有限时间H∞滤波的Riccati方程和滤波方程分别为非线性矩阵微分方程和线性变系 数微分方程,而且Riccati微分方程解的存在性还依赖于参数 γ-2,因此求这些方程的数值解一 般比较困难.按照结构力学与最优控制的模拟关系,Riccati方程解存在的临界参数 γ-2cr对应于 广义Rayleigh商的一阶本征值.因此可以用精细积分法结合扩展的Wittrick-Williams(W-W) 算法计算 γ-2cr .并求解Ricclati方程,滤波微分方程的解也可以由精细积分法计算.     

具有状态依赖脉冲控制的无公害害虫管理模型

研究具有状态依赖脉冲控制的无公害害虫管理模型,利用微分方程几何理论中后继函数法得到系统阶一周期解存在的充分条件,证明该周期解是轨道渐近稳定的,并对系统进行了数值模拟。     

电站并联运行的不对称非线性振荡

本文给出一个结构不完全对称并联电网的等价定理,它把双输入双输出非线性耦合的微分方程组等价为单输入单输出的非线性微分方程,然后用渐近方法和谐波线性化方法求其一次近似解,得到一些新的物理性质,有助于合理选择电网结构,以提高其结构稳定性.     

插值有理Bézier渐近四边形的有理Bézier曲面

为构造封闭的曲线为有理Bézier曲面的边界渐近线,给出封闭四边曲线为渐近四边形的条件,并提出插值该四边形的曲面构造方法.首先在给定角点数据的前提下构造优化的n次有理Bézier渐近四边形;然后利用该四边形和曲面在四边形上的切矢确定曲面沿边界的两排控制顶点和权;最后极小化曲面薄板能量函数确定剩余自由的控制顶点,进而构造出光滑的双5n–7次有理Bézier插值曲面.实例展示边界曲线为有理3,4,5次时曲面的构造结果,以及边界曲线含有直线或者拐点的情况,表明该方法是可行的.     

有理Bézier单元求解参数曲面上Laplace-Beltrami方程

用有限元法数值求解时,定义在流形曲面上的偏微分方程的数值解精度会因为传统多边形单元的几何逼近误差而严重降低,为此提出基于有理Bernstein多项式的几何精确有限元法.首先插入重复节点从NURBS曲面直接生成有理Bézier单元,这一过程保持原有几何不变;然后通过Galerkin法建立参数曲面上包含Laplace-Beltrami微分算子的二阶椭圆偏微分方程的等效弱形式;针对Bernstein基函数的非插值性,通过配点法施加Dirichlet类型的边界约束,得到最优收敛的离散格式.数值算例结果表明,该方法能有效地减少网格离散误差,提高分析结果精度.     

对线性系统状态转移矩阵的讨论

状态转移矩阵是现代控制理论的重要概念,在线性控制系统的运动分析中起着重要的作用.分别对连续时间线性时变系统、离散时间线性定常系统以及离散时间线性时变系统的状态转移矩阵进行了研究.根据常微分方程和差分方程解的唯一性,得到了判断矩阵函数是某一线性系统状态转移矩阵的充分条件,并求出了其对应的系统矩阵.实践证明了结论的正确性.     

偏微分方程的MATLAB数值解法及可视化

偏微分方程的数值解法在数值分析中占有很重要的地位,很多科学技术问题的数值计算包括了偏微分方程的数值解问题。在学习初等函数时,总是先画出它们的图形,因为图形能帮助了解函数的性质。而对于偏微分方程,画出它们的图形并不容易,尤其是没有解析解的偏微分方程,画图就显得更加不容易了。为了从偏微分方程的数学表达式中看出其所表达的图形、函数值与自变量之间的关系,通过MATLAB编程,数值求解了泊松方程,并将其结果可视化,给出了解析解与数值解的误差。     

用余弦微分求积法数值求解KdV—Burgers方程

采用余弦微分求积法（CDQM）对（1＋1）维非线性KdV—Burgers方程进行了数值求解．结果表明,所得数值解与方程的精确解相比具有明显的高精度且稳定性高,相对于其他常用方法,且公式简单,使用方便;计算量小,时间复杂性好．     

Causes of instabilities in numerial integration techniques

This paper presents the causes of instabilities which arise during the numerical solution of ordinary differential equations. Using the numerical integration routines presently available, one actually approximates the differential equation by a difference equation. If the difference equation is of higher order than the original differential equation, the approximate solution contains extraneous solutions which are not at all related to the true solution. It is the behavior of these extraneous solutions that one is concerned with in a stability analysis.     

Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method

During the past few decades, the idea of using differential quadrature methods for numerical solutions of partial differential equations (PDEs) has received much attention throughout the scientific community. In this article, we proposed a numerical technique based on polynomial differential quadrature method (PDQM) to find the numerical solutions of two-dimensional sine-Gordon equation with Neumann boundary conditions. The PDQM reduced the problem into a system of second-order linear differential equations. Then, the obtained system is changed into a system of ordinary differential equations and lastly, RK4 method is used to solve the obtained system. Numerical results are obtained for various cases involving line and ring solitons. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions that exist in literature. It is shown that the technique is easy to apply for multidimensional problems.     

Wick类型的随机广义Kdv方程的精确解

在Kondratiev分布空间(S)-1中通过埃尔米特变换和Painleve′分析导出了Wick-类型的随机广义Kdv方程的Backlund变换,并且把Wick-类型的随机广义Kdv方程变成广义系数Kdv-方程,再利用Backlund变换求出广义系数Kdv方程的精确解,最后通过埃尔米特逆变换求出随机广义Kdv方程在系数取不同白色噪音泛函条件下的精确解.     

A localized approach for the method of approximate particular solutions

The method of approximate particular solutions (MAPS) has been recently developed to solve various types of partial differential equations. In the MAPS, radial basis functions play an important role in approximating the forcing term. Coupled with the concept of particular solutions and radial basis functions, a simple and effective numerical method for solving a large class of partial differential equations can be achieved. One of the difficulties of globally applying MAPS is that this method results in a large dense matrix which in turn severely restricts the number of interpolation points, thereby affecting our ability to solve large-scale science and engineering problems.In this paper we develop a localized scheme for the method of approximate particular solutions (LMAPS). The new localized approach allows the use of a small neighborhood of points to find the approximate solution of the given partial differential equation. In this paper, this local numerical scheme is used for solving large-scale problems, up to one million interpolation points. Three numerical examples in two-dimensions are used to validate the proposed numerical scheme.     

Convergence of the Euler–Maruyama method for stochastic differential equations with Markovian switching

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Most of SDEwMSs do not have explicit solutions so it is important to have numerical solutions. It is surprising that there are not any numerical methods established for SDEwMSs yet, although the numerical methods for stochastic differential equations (SDEs) have been well studied. The main aim of this paper is to develop a numerical scheme for SDEwMSs and estimate the error between the numerical and exact solutions. This is the first paper in this direction and the emphasis lies on the error analysis.     

Similarity solutions for axisymmetric plane radial power law fluid flows through a porous medium

This paper investigates the unsteady flow of non-Newtonian fluids of power low behavior through a porous medium in a plane radial geometry. The equation governing the flow is a nonlinear parabolic partial differential equation with a source term whose solution satisfies certain fixed and moving boundary conditions. The attention is focused on the finding of similarity solution when the fixed boundary condition and the source term satisfy certain restrictions. In this case similarity transformations are determined and the resulting ordinary differential equations are deduced. For shear thinning fluids the existence of a pressure disturbance front moving with finite velocity is shown and expression for its location as a function of time is determined. The solutions in closed form have been given for certain particular cases where the resulting differential equations can be analytically solved. A numerical procedure has also been presented.     

Using reproducing kernel for solving a class of time-fractional telegraph equation with initial value conditions

Today, most of the real physical world problems can be best modelled with fractional telegraph equation. Besides modelling, the solution techniques and their reliability are the most important. Therefore, high accuracy solutions are always needed. As we all know, reproducing kernel method (RKM) has been successfully presented for solving various ordinary differential equations. However, the numerical results are not perfectly satisfactory when we directly use the traditional RKM for solving fractional partial differential equation. The aim of this paper is to fill this gap. In this paper, a new method is provided for solving fractional telegraph equation in the reproducing kernel space by piecewise technique, which can obtain more accurate solution than traditional method. Three experiments are given to demonstrate the effectiveness of the present method.     

Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme

This paper proposes a split cosine scheme for simulating solitary solutions of the sine-Gordon equation in two dimensions, as it arises, for instance, in rectangular large-area Josephson junctions. The dispersive nonlinear partial differential equation allows for soliton-type solutions, a ubiquitous phenomenon in a large variety of physical problems. The semidiscretization approach first leads to a system of second-order nonlinear ordinary differential equations. The system is then approximated by a nonlinear recurrence relation which involves a cosine function. The numerical solution of the system is obtained via a further application of a sequential splitting in a linearly implicit manner that avoids solving the nonlinear scheme at each time step and allows an efficient implementation of the simulation in a locally one-dimensional fashion. The new method has potential applications in further multi-dimensional nonlinear wave simulations. Rigorous analysis is given for the numerical stability. Numerical demonstrations for colliding circular solitons are given.