Nonstandard perturbation approximation and travelling wave solutions of nonlinear reaction diffusion equations

This paper deals with the construction of a nonstandard numerical method to compute the travelling wave solutions of nonlinear reaction diffusion equations at high wave speeds. Related general properties are studied using the perturbation approximation. At high wave speed the perturbation parameter approaches to zero and the problem exhibits a multiscale character. That is, there are thin layers where the solution varies rapidly, while away from these layers the solution behaves regularly and varies slowly. Most of the conventional methods fail to capture this layer behavior. Thus, the quest for some new numerical techniques that may handle the travelling wave solutions at high wave speeds earns relevance. In this paper, one such parameter robust nonstandard numerical scheme is constructed, in the sense that its numerical solution converges in the maximum norm to the exact solution uniformly well for all finite wave speeds. To overcome the difficulty due to the nonlinearity, the problem is linearized using the quasilinearization process followed by nonstandard finite difference discretization. An extensive amount of analysis is carried out which uses a suitable decomposition of the error into smooth and singular component and a comparison principle combined with appropriate barrier functions. The error estimates are obtained, which ensures uniform convergence of the method. A set of numerical experiment is carried out in support of the predicted theory that validates computationally the theoretical results. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

一个具有时滞和功能反应的非自治阶段结构的捕食模型

研究一个具有时滞和功能反应的非自治阶段结构的捕食模型,其中食饵种群分两个阶段——幼年和成年,且捕食种群只捕成年食饵.通过利用微分不等式、比较定理和构造适当的Lyapunov泛函,得到该模型的永久持续生存和绝灭的充分条件,同时也得到了周期系统存在唯一全局渐近稳定的周期解的充分条件.     

双醛葡甘聚糖溶液的荧光光谱特性研究

研究了双醛葡甘聚糖水溶液在不同紫外光激发下的荧光光谱特性。结果表明,在220～320 nm紫外光激发下,双醛葡甘聚糖分子环结构中C—O—C醚键单元的氧未共价成键电子吸收激发光子产生n→σ＊跃迁,醛基单元中C=O氧未共价成键电子吸收激发光子产生n→π＊跃迁,最大发射波长分别在425 nm和465 nm左右;溶剂通过氢键等相互作用对双醛葡甘聚糖溶液荧光光谱产生显著影响。研究结果对探索葡甘聚糖水凝胶给药载体与小分子药物的相互作用机理及其释药行为具有指导意义。     

Dead cores of singular Dirichlet boundary value problems with ϕ-Laplacian

The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem (ϕ(u′))′ = λf(t, u, u′), u(0) = u(T) = A. Here λ is the positive parameter, A > 0, f is singular at the value 0 of its first phase variable and may be singular at the value A of its first and at the value 0 of its second phase variable. This work was supported by grant no. A100190703 of the Grant Agency of the Academy of Sciences of the Czech Republic and by the Council of Czech Government MSM 6198959214.     

Triple positive solutions for a class of -point dynamic equations on time scales with -Laplacian

This paper deals with a class of second-order nonlinear m-point dynamic equation on time scales with one-dimensional p-Laplacian. Using a fixed point theorem for operators on a cone, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The interesting point is that the nonlinear term f is involved with the first-order delta derivative explicitly. Meanwhile, an example is worked out to demonstrate the main results.     

带p-Laplacian算子的奇异脉冲微分方程正解的存在性

运用不动点指数理论,我们研究一类带p-Laplacian算子的奇异脉冲微分方程两点边值问题正解的存在性.     

在共振点附近的一类二阶泛函微分方程的解析解

在复域C内研究一类包含未知函数迭代的二阶微分方程x″(z)=G(z,x(z),x~2(z),…,x~m(z))解析解的存在性.通过Schr(?)der变换,即x(z)=y(αy~(-1)(z)),把这类方程转化为一种不含未知函数迭代的泛函微分方程α~2y″(αz)y″(z)-αy′(αz)y″(z)= (y′(z))~3G(y(z),y(αz),…,y(α~mz)),并给出它的局部可逆解析解.本文不仅讨论了双曲型情形0<|α|<1和共振的情形(α是一个单位根),而且还在Brjuno条件下讨论了共振点附近的情形(即单位根附近).     

一类广义Tjon-Wu方程的渐近稳定性

研究Boltzmann方程的一个动力学模型:Tjon-Wu方程的更一般的形式.我们证明了在L_(1.1)范数意义下方程的稳态解的渐近稳定性.     

Existence of positive periodic solutions for two kinds of neutral functional differential equations

In this work, we deal with a new existence theory for positive periodic solutions for two kinds of neutral functional differential equations by employing the Krasnoselskii fixed-point theorem. Applying our results to various mathematical models we improve some previous results.     

Transient entanglements and clusters in dilute polymer solutions

A convenient method to monitor polymer dissolution is to measure the pressure drop created by passing a polymer solution through a capillary constriction rheometer. In this work, we studied the dissolution of polyethylene oxide (PEO) and cationic starch (C‐starch). We found that for freshly dissolved and entangled PEO, the main contribution to the overall pressure drop is due to the contraction and expansion of PEO entanglements at the entrance and exit of the capillary, and that the friction in the capillary plays a minor role. On the other hand, for well‐dissolved PEO, because of the absence of PEO entanglements, the loss of pressure is mainly due to friction. At high velocities the contraction and expansion coefficient for freshly dissolved PEO was more than 20 times higher than for well‐dissolved PEO, resulting in a three times higher overall pressure drop. C‐starch consists of amylopectin (～ 85%) and amylose and is known to contain clusters when freshly dissolved, likely formed from the globular amylopectin molecules. For C‐starch, the main contribution to the overall pressure drop is due to friction. Entrance and exit effects contribute only 10% to the overall pressure drop, which might be due to the linear amylose molecules in C‐starch. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 253–262, 2008