不连续二阶非线性微分方程的周期边值问题

利用上下解方法和不动点定理,给出了含导数项的不连续二阶非线性微分方程周期边值问题的极解。     

一类带有扩散和B-D反应项的病毒模型的稳定性分析

本文研究了齐次Neumann边界条件下带有扩散和B-D反应项病毒模型的平衡解渐近稳定性.利用弱耦合抛物不等式组的最大值原理,给出了模型解的先验估计.利用赫尔维茨(Hurwitz)定理,分析了平衡解的局部渐近稳定性.结果表明:当基本再生数大于1时,地方病平衡态局部渐近稳定;当基本再生数小于1时,无病平衡态局部渐近稳定.同时,利用构造上下解及其单调迭代序列的方法证明了无病平衡解的全局渐近稳定性,该结果表明:当控制细胞生成率或者感染率或者感染细胞裂解率充分小时,无病平衡解的全局渐近稳定.     

二阶方程组解的存在唯一性

本文在抽象空间中研究了不连续二阶常微分方程组解的存在唯一性,利用单调迭代方法和上下解方法证明了方程组的唯一解可以由迭代序列的一致极限得到,并给出了逼近解的迭代序列的误差估计式。     

一阶泛函微分方程解的存在唯一性

利用了上下解方法研究一阶泛函微分方程Cauchy问题的解的存在唯一性问题，减弱了通常的Lip-schitz条件。     

一类Leslie-Gower捕食食饵模型的分歧

本文研究了一类具有扩散的Leslie-Gower模型.利用谱分析和稳定性理论得到了两个正常数平衡态的局部稳定性;利用最大值原理、Harnack不等式和能量积分的方法得到了正稳态解的上下界估计和非常数正解的存在性;利用单特征值分歧理论研究了系统发自两个正常数平衡态的解分支,得到了非常数正解的存在性;利用Hopf分歧理论,得到了在平衡解处Hopf分歧的存在性.     

时标上脉冲动力方程周期边值问题的拟线性化方法

本文研究了一类时标上脉冲动力方程周期边值问题解的收敛性问题.利用时标上一阶脉冲动力不等式、上下解和单调迭代技巧证明了该问题解的一致收敛性结果,并进一步采用拟线性化方法和分析技巧获得了该方程在周期边值条件下两个逼近解序列高阶收敛的充分性判据.本文所得结果发展了时标上动力方程定性理论的结果.     

一类恒化器竞争模型共存态的全局分歧北大核心CSCD

研究了一类带Ivlev型反应函数的非均匀恒化器竞争模型的全局分歧.利用最大值原理获得了共存解的先验估计,借助于特征值理论、上下解方法得到了共存解存在的必要条件,采用局部分歧理论构造了共存解的局部分支,并运用全局分歧理论证明了共存解的局部分支可延拓为全局分支.结果表明该全局分支连接了模型的两半平凡解分支.从生物学角度看,当两竞争物种的最大生长率满足一定条件时,两物种可以共存.     

带时滞项的中立型脉冲微分方程的周期边值问题

脉冲微分方程是模拟控制理论、物理学、化学、生物技术、工业机器人等方面的一些过程和现象的一种非常好的模型.本文研究了带时滞项的中立型脉冲微分方程的周期边值问题的极小值与极大值解的存在性.首先引入了方程新的上下解概念,然后发展了一个脉冲不等式.利用它们和单调迭代法,获得了两个新的比较原理,并利用线性化的方法,进一步建立了该...     

四阶奇异边值问题的正解

利用了上下解方法和不动点定理，得到了一类四阶奇异边值问题正解的存在性。     

带有扩散项具有阶段结构的两种群捕食-食饵系统近似波前解的存在性

本文研究一类带有扩散项具有阶段结构的两种群捕食-食饵系统近似波前解的存在性.通过线性化方法,首先分析了两种群时滞反应扩散系统平衡点的渐近稳定性.然后,把一致逼近方法与上下解方法相耦合,通过构造满足一定光滑性的上下解,证明了当波速足够大时,带有扩散项具有阶段结构的两种群捕食-食饵系统近似波前解的存在性.在一定条件下,解决...     

非线性分数阶耦合泛函微分方程组边值问题的可解性

由于运动速度是有限的，因此在信号传输等过程中时滞现象往往是不可避免的。分数阶泛函微分方程是研究时滞系统运动规律的重要模型，当系统中具有两个或多个状态变量且这些状态变量相互作用时，常常运用耦合微分方程组来刻画。对一类具有 Riemann-Liouville 分数阶导数的非线性时滞耦合泛函微分方程组边值问题正解的存在唯一性进行了研究。首先，根据方程与边界条件的特点，建立了比较定理，构造了上解与下解的单调序列，并确定了上下解的关系。运用上下解的方法建立并证明了边值问题正解的存在性定理，同时得到了正解的取值范围。然后，利用迭代技术建立并证明了边值问题正解的存在唯一性定理。最后，给出了具体例子用于说明所得主要结论的适应性与广泛性。     

具有多时滞和广义扩散的N-种群竞争反馈控制系统的持久性

本文讨论了一类具有αi类功能性反应函数和广义扩散的n种群竞争反馈控制生态系统的持续生存性.利用比较原理,得到了系统的所有正解最终有界的条件.通过构造持久生存函数,给出了系统一致持久生存的充分条件,并导出了系统的持久生存域.最后,通过建立具体模型说明所得结果的可实现性.研究结果表明,时滞不影响系统的持续生存性,通过适当控制扩散率,可使系统中的各个种群长期共存.     

Symbolic homotopy construction

The classical Theorem of Bézout yields an upper bound for the number of finite solutions to a given polynomial system, but is very often too large to be useful for the construction of a start system, for the solution of a polynomial system by means of homotopy continuation. The BKK bound gives a much lower upper bound for the number of solutions, but unfortunately, constructing a start system based on this bound seems as hard as solving the original given polynomial system. This paper presents a way for computing an upper bound together with the construction of a start system. The first computation is performed symbolically. Due to this symbolic computation, the constructed start system can be solved numerically more efficiently. The paper generalizes current approaches for homotopy construction towards the BKK bound.     

A novel singular node‐based smoothed finite element method (NS‐FEM) for upper bound solutions of fracture problems

It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node‐based smoothed finite element method (NS‐FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS‐FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five‐node singular crack tip element is used within the framework of NS‐FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix‐mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS‐FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

非线性分数阶积分微分方程边值问题正解的存在性

研究一类具有两个分数阶导数项的非线性分数阶积分微分方程积分边值问题。首先将原问题转化为只有一个导数项的等价形式,通过定义等价问题的上下解,再利用单调迭代技术建立了原问题正解的存在性与唯一性定理,给出了求其唯一正解的迭代格式和误差估计。最后给出实例说明所得结论的有效性和适用性。     

Nonlinear steady two-layer interfacial flow about a submerged point vortex

Two-dimensional, two-layer steady interfacial flow about a point vortex is studied in a uniform stream for each layer. The upper layer is of finite depth with a rigid lid on the upper surface, and the depth of the lower layer is assumed infinite. The point vortex is located in lower-layer fluid. We study this problem using not only a linear analytical method but also a nonlinear numerical method. A linear solution is derived in terms of a complex exponential integral function. The fully nonlinear problem is formulated by an integro-differential equation system. The equation system is solved using Newton’s method to determine the unknown steady interfacial surface. The numerical results of the downstream wave are provided by a linear solution and fully nonlinear solution. A comparison between linear solutions and nonlinear solutions shows that the nonlinear effect is apparent when the vortex strength increases. The effects of point vortex strengths, Froude numbers, and density ratios on the amplitudes of the downstream waves are studied. We analyze the effects of point vortex strengths, Froude numbers, and density ratios on the wavelengths of the downstream waves.     

The application of a variational finite element method to problems in fluid dynamics

The finite element method is applied to the analysis of a two-dimensional steady flow around a box girder in uniform flow. The local potential principle for an incompressible viscous fluid flow is introduced as a basis for establishing finite element models and a determination is made as to whether or not approximate solutions obtained by the use of the principle give upper or lower bounds of the solution. The boundaries are set so as to be appropriately far from the girder and the velocities are specified at the upstream, upper boundary and the lower boundary and the total stresses are specified at the downstream boundary. The system of non-linear equations is solved by the standard Newton–Raphson method. Integrating the pressures and viscous stresses acting on the surface of the girder, the coefficients of drag, lift and pitching moment are numerically obtained for some Reynolds numbers. The wind tunnel test has been the only means to determine these coefficients for a body with irregular shape. The finite element method presents a new powerful procedure to determine these coefficients.