两类破碎群体平衡方程接受的李群及群不变解

针对难找到破碎群体平衡方程的精确解和解析方法缺乏的问题,研究两类积分-偏微分方程(破碎群体平衡方程)接受的李群、群不变解、约化积分-常微分方程及精确解.首先采用伸缩变换李群分析方法探寻积分-偏微分方程接受的李群.其次将积分-偏微分方程转化为纯偏微分方程,运用经典李群分析方法计算纯偏微分方程接受的李群.然后利用改进了的李群分析方法结合伸缩变换群和经典李群分析方法获得的结果确定积分-偏微分方程接受的李群.最后找到了积分-偏微分方程接受的李群,给出了积分-偏微分方程的约化积分-常微分方程、群不变解及显式精确解,分析了部分解的动力学行为性质及特征.     

非线性演化方程的孤立波解

用齐次平衡原则和辅助微分方程方法得到了6个重要的n次非线性演化方程的孤立波解.辅助微分方程方法的主要思想是借助简单的可解微分方程的解去构造复杂的非线性演化方程的行进波解.这里简单的可解微分方程称为辅助微分方程.本文使用的辅助方程有双曲正割幂型解或双曲正切幂型解.     

无阻尼振动方程解的稳定性

在无干扰力的环境中,定性分析无阻尼振动方程解的稳定性,可归结为下述几个问题:牛顿第二运动定律应用于振动建模描述力与运动的关系,线性化方程是求解微分方程的有效方法;单摆振动的等时性与非等时性特征表明,微分方程的解不仅决定于方程本身,而且也决定于解的初值;微分方程定性理论,特别是李雅普诺夫第二方法,是研究非线性微分方程解的稳定性的有效手段;如何构造李雅普诺夫函数,至今仍是一个吸引人的研究课题.     

全微分方程与积分因子定义的修正

一个方程是否为全微分方程,或者一个方程是否有积分因子,都与方程的定义区域密切相关.本文纠正了通行教材中全微分方程和积分因子的不严谨的定义,并修正了全微分方程判定定理和方程仅有依赖于单个变量积分因子的判定定理.     

三维半导体问题的迎风有限体积格式

半导体器件的瞬时状态由包含三个拟线性偏微分方程所组成的方程组的初边值问题来描述.其中电子位势方程是椭圆型的,电子和空穴浓度方程是对流扩散型的.作者对三维半导体模型问题采用四面体网格上的有限体积元方法进行逼近,具体地,对电子位势方程采用一次元有限体积法来逼近,对电子浓度和空穴浓度方程采用迎风有限体积方法来逼近,并进行了详细的理论分析,得到了O(h+\Delta t)阶的L^2模误差估计结果.     

求解Fisher型方程的混合Legendre-球面调和谱方法

主要研究两同心球所界球形区域上偏微分方程的谱方法,建立了与区域形状相适应的混合Legendre-球面调和正交逼近的部分结果,在此基础上提出了数值求解两同心球所界球形区域上Fisher型方程的混合Legendre-球面调和谱格式,并分别给出了格式的收敛性及相关的数值结果.     

Kadomtsev-Petviashvili方程的精确解

齐次平衡法是把非线性偏微分方程转换成带约束条件的线性偏微分方程的一种很好的方法 .本文在齐次平衡法的基础上具体讨论了KP方程的精确解 ,包括孤波解 ,一般的行波解 ,有理函数解和一种新类型的解 .     

A Well-Balanced HLL Scheme for Hyperbolic Conservation Systems With Source Terms北大核心CSCD

针对含源项的双曲守恒方程给出了一种新的有限体积格式.经典的有限体积格式不能正确地模拟对流通量项和外力之间的平衡所产生的动力学问题.为解决这个问题,仿照经典的HLL近似Riemann求解器设计思路设计了含源项的近似Riemann求解器.针对含重力源项的一维流体Euler方程和理想磁流体方程,通过对通量计算格式的修正得到了保平衡HLL格式（WB-HLL）,并给出了保平衡的证明.针对一维Euler方程和理想磁流体给出了两个算例,比较了传统HLL格式和提出的WB-HLL格式的计算精度.计算结果表明,WB-HLL格式精度更高,收敛更快.     

横观各向同性压电材料的基本解

推导得到一组在体积力作用下压电材料平衡方程的一般解,对横观各向同性压电材料,利用一般解结合体积势理论及构造一类调和函数的方法,得到了无限体在集中力和点电荷作用下的位移和电势的有限形式的表达式,从而给出了边界元法中可用的基本解.     

高维旋转曲面

本文考虑欧氏空间中一种余一维的高维旋转曲面,通过发展出一种全新的复合映射、维数分解与分块矩阵递推法,我们系统性地研究了同它的面积和曲率有关的一系列问题.当母函数是多元函数时,这种高维旋转曲面的概念尚属首次提出.我们给出了这种高维旋转曲面的面积公式以及它的一些简单应用.我们发现:在任一直径方向上,单位球面的面积分布和低一维单位球体的体积分布完全相同,并且当维数趋于无穷时它们的密度函数的极限都是狄拉克函数.通过研究相应面积泛函的变分问题,我们得到了所谓的极小旋转曲面方程.我们证明了:满足极小旋转曲面方程的母函数对应的旋转曲面的平均曲率等于零.这种极小旋转曲面方程推广了传统的极小曲面方程,并且为非参数极小曲面理论提供了新的更一般的研究框架;通过计算径向对称解对应的常微分方程,我们研究了它的一些简单的特解.我们也简单讨论了相应的预定平均曲率和预定高斯曲率问题.     

On Some Integral Transformations and Their Application to the Solution of Boundary-Value Problems in Mathematical Physics

We obtain a formula for the expansion of an arbitrary function in a series in the eigenfunctions of the Sturm–Liouville boundary-value problem for the differential equation of cone functions. On the basis of this result, we derive a series of integral transformations (including well-known ones) and inversion formulas for them. We apply these formulas to the solution of initial boundary-value problems in the theory of heat conduction for circular hollow cones truncated by spherical surfaces.     

厚球壳与实心球轴对称问题的一般解

本文试图从更一般的三维问题基本方程出发研究任意厚球壳与实心球的轴对称问题.对于受任意轴对称载荷的厚球壳和实心球体,文中运用加权残值法给出了以Legendre级数表示的一般解.     

A new lattice Boltzmann model for the laplace equation

In this paper, a new lattice Boltzmann equation which is independent of time is proposed. Based on the new lattice Boltzmann equation, some steady problems can be modeled by the lattice Boltzmann method. In the further study, the Laplace equation is investigated with the method of the higher-order moment of equilibrium distribution functions and a series of partial differential equations in different space scales. The numerical results show that the new method is effective.     

Mathieu functions and coulomb spheroidal functions in the electrostatic probe theory

A spherical probe placed in a slowly moving collisional plasma with a large Debye length λ_D → ∞ is considered. The partial differential equation describing the electron concentration around the probe is reduced to two ordinary differential equations, namely, to the equation for Coulomb spheroidal functions and Mathieu’s modified equation with the parameter a of the latter related to the eigenvalue λ of the former by the relation a = λ + 1/4. It is shown that the solutions of Mathieu’s equation are Mathieu functions of half-integer order, which are expressed as series in terms of spherical Bessel functions and series of products of Bessel functions. These Mathieu functions are numerically constructed for Mathieu’s modified and usual equations.     

Dynamical systems and variational inequalities

The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modeling, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equilibrium problems.     

求解非线性互补问题的微分方程方法

本文构造了一种求解非线性互补问题的微分方程方法.在一定条件下,证明了微分方程系统的平衡点是非线性互补问题的解并且基于一般微分方程系统的数值积分建立了一个数值算法.在适当的条件下,证明了此算法产生的序列解是收敛的.本文最后给出了数值结果,该结果表明了此微分方程方法的有效性.     

时滞抛物型与双曲型微分方程解的振动性的讨论

对一类线性以及非线性抛物型时滞微分方程的解在第一或第二边值条件下解振动的充分必要条件进行了讨论,给出了解振动的一些结论.并且对一类线性以及带强迫项的非线性双曲型时滞微分方程的解在第一或第二边值条件下解振动的充分必要条件进行了讨论,也给出了一些结论.     

A new method for solving the eikonal equation in the spherical computing domain

In order to overcome the problem of singularities and nonuniform grids arising when solving eikonal equation in spherical coordinate systems, a spherical Cartesian coordinate system is defined and the Hamiltonian form of the eikonal equation according to this coordinate system is given. A modified velocity function that can transform spherical coordinate system–based eikonal equation into ones based on a spherical Cartesian coordinate system is deduced by using a differential geometric method where a layered distribution of the velocity function is assumed. After comparing the results of using this approach with the traditional method of solving eikonal equation based on a spherical coordinate system, the viability of the transformation to a spherical Cartesian coordinate system based on a modified velocity function is proven. Despite the assumption of a layered distribution of the velocity function, it is also proven that the method will hold for a velocity function under any three-dimensional distribution. The new method overcomes problems present in traditional approaches and opens up a new way of solving eikonal equation in a spherical computational domain.     

A plasmastatic model of the galathea-belt magnetic trap

The mathematical apparatus of plasmastatics, which includes the MHD equilibrium equation and steady-state Maxwell equations, is reduced, in two-dimensional problems arising due to symmetry, to a single scalar second-order elliptic equation with a nonlinear right-hand side known as the Grad-Shafranov equation. In this paper, we numerically solve a series of boundary value problems for this equation that model equilibrium plasma configurations in the magnetic field of the belt-like galathea trap in a cylinder with two plasma embedded conductors. The mathematical model is outlined, the results of calculations of the magnetic field and plasma pressure in the cylinder depending on the parameters of the problem are presented, and the main integral characteristics of the trap are calculated. The existence and uniqueness of the solution is discussed; the limiting values of the maximal pressure at which there exists a solution of the equilibrium problem are found.     

On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces

We present some conditions for the asymptotic equilibrium of nonlinear differential equations and, in particular, a linear inhomogeneous equation in Banach spaces. We also discuss analogous problems for a linear equation with unbounded operator. Some obtained results are applied to problems of asymptotic equivalence. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 626–635, May, 2008.