拟线性抛物型方程奇异摄动问题的数值解法

本文讨论拟线性抛物型方程奇异摄动问题的差分解法,在非均匀网格上建立了线性三层差分格式,并证明了在离散的L₂范数意义下格式的一致收敛性,最后给出了一些数值例子.     

Conditions for the Time Global Existence and Nonexistence of a Compact Support of Solutions to the Cauchy Problem for Quasilinear Degenerate Parabolic Equations

We study a quasilinear degenerate parabolic equation with a nonhomogeneous density. We establish that depending on the behavior of the density at infinity either a solution to the Cauchy problem possesses the property that the speed of propagation of perturbations is finite or the support blows up in finite time.     

一类拟线性抛物系统解的整体存在性

采用非线性分析的方法 ,在任意带光滑边界的区域内 ,研究一类带交错扩散影响的拟线性抛物系统 ,证明了解的整体存在性 .     

一阶椭圆型复方程组Riemann－Hilbert边值问题

本文以解析函数的边值问题Ｂ的解的存在性为基础，根据它们的先验估计式及利用参数开拓法，导出了满足条件Ｃ的多个复变量的一阶拟线性椭圆型复方程组的Ｒｉｅｍａｎｎ－Ｈｉｌｂｅｒｔ边值问题的可解条件，并给出了解的积分表达式．     

一类带一般交错扩散影响的拟线性抛物系统解的整体存在性

采用对偶性技巧及 Holder不等式 ,在 R2上的任意带光滑边界的区域内 ,研究一类带一般交错扩散影响的拟线性抛物系统 ,证明了解的整体存在性 .     

Volteera型积分微分方程组边值问题的奇摄动

该文研究二阶积分微分方程组边值问题奇摄动，在适当的条件下，利用渐近分析方法和对角化技巧，还得解的存在性和给出解的渐近展开式与相应的余项估计．然后，应用这些结果到三阶常微分方程组边值问题的奇摄动，最后也得到解的一致有效的渐近展开式．     

Cauchy Problem for a System of Equations of Ultraparabolic Type

This paper is devoted to the proof of the existence of a solution of the Cauchy problem for a system of equations of ultraparabolic type.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 768–774.Original Russian Text Copyright ©2005 by S. A. Tersenov.     

Time asymptotic behavior of the solution to a quasilinear parabolic equation

The time asymptotic behavior of the solution to the Cauchy problem for a quasilinear parabolic equation is analyzed. Such problems are encountered, for example, in gas dynamics and transport flow simulation. A.M. Il’in and O.A. Oleinik’s well-known results are extended to a wider class of equations in which the time derivative of the unknown function is multiplied by a fixed-sign function of the former. The results have found applications in mathematical economics.     

On the effects of the Bohm potential on a macroscopic system of self-interacting particles

We consider a nonstationary macroscopic system of self-interacting particles with an additional potential, the so called Bohm potential. We study the existence of nonnegative global solutions to the system of equations and allude to the differences to results obtained for classical models. The problem is considered on a bounded domain up to three spatial dimensions, subject to initial and Neumann boundary conditions for the particle density, and the Dirichlet boundary condition for the self-interacting potential. Moreover, the initial datum is only assumed to be nonnegative and to satisfy a weak integrability condition.     

Probabilistic interpretation of a system of quasilinear parabolic PDEs

Using a forward–backward stochastic differential equations (FBSDE) associated to a transmutation process driven by a finite sequence of Poisson processes, we obtain a probabilistic interpretation for a non-degenerate system of quasilinear parabolic partial differential equations (PDEs). The novetly is that the linear second order differential operator is different on each line of the system.     

一类半线性抛物型方程组解的爆破速率估计

考虑一类带有齐次Dirichlet边界条件且反应项分别为指数形式和幂函数形式的半线性抛物型方程组,利用比较原理得到了方程解爆破的充分条件,由数学分析原理和最大值原理得到了爆破解的爆破速率估计.     

一阶系统非线性边值问题的奇摄动*

本文应用比较定理研究了一类非线性边界条件的向量非线性奇摄动问题εx='f(t,x,y,e)εy'=g(t,x,y,ε)x(0)=A(ξ₁,ξ₂,x(1)-x(0),y(1)-y(0),ε)y(0)=B(ξ₁,ξ,x(1)-x(0),y(1)-y(0),ε)这里ξ₁,ξ₂为ε的函数。0<ε<<1,在适当的条件下,作出了任意次精度的渐近展式。并得出余项估计。     

Analysis of a Free Boundary Problem Modeling Tumor Growth

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions： R = R（t）, the radius of the tumor, and u = u（r, t）, the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R（t）, t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞.     

含积分算子拟线性系统边值问题的奇摄动

研究奇摄动积分微分方程组边值问题εy"＝f(x,y,Ty,ε)y′++g(x,y,Ty,ε);y(0,ε)＝A(ε),y(1,ε)=B(ε)其中y、g、A和B均为n维向量函数,f是n×n矩阵函数,(Ty)(x)=∫xK(x,s,y(sε),ε)ds在一定假设条件下,利用对角化技巧和逐步逼近法证明解的存在,并给出解的直到0(εN+1)的渐近展开式.     

Local existence inC b 0, 1 and blow-up of the solutions of the Cauchy Problem for a quasilinear hyperbolic system with a singular source term

In this paper we consider the Cauchy problem for the hyperbolic system

Well-Definedness of the Cauchy Problem for the One-Dimensional Equations of Viscous Heat Conducting Gas with Initial Data from Lebesgue Spaces

Mathematical Notes -     

拟线性常微分方程组边值问题解的估计

本文研究拟线性常微分方程组边值问题x′=f(t,x,y,ε),x(0,ε)=A(ε) εy″=g(t,x,y,ε)y′+h(t,x,y,ε) y(0,ε)=B(ε),y(1,ε)=C(ε)的奇摄动。其中x,f,y,h,A,B和C均属于Rn,g是n×n矩阵函数。在适当的条件下,利用对角化技巧和不动点定理证明解的存在,并估计了余项.     

基于机场货运预测的GM(1,1)建模过程问题分析

运用灰色系统GM(1,1)模型进行预测时,模拟精度和预测精度是关注的焦点.对原始数据构造缓冲算子和进行完整的前期检验,运用等维灰数递补的动态建模方法,针对误差建立残差修正模型,都可以提高预测的可信度.通过归纳建模各阶段可能出现的典型问题以及解决方法,以期对建立优化模型有所帮助.最后建立GM(1,1)模型预测天津滨海机场货邮吞吐量.     

一个游戏难题的推广、建模及求解

推广了RPG游戏中的一个难题,建立了相应的数学模型,给出了完善的解决方案,深化了现行的相关结果.     

2010年全国研究生数模竞赛D题的评阅综述

2010年全国研究生数学建模竞赛D题的素材取自一项科研项目.将本题建为何种模型取决于建模人的数学基础和对本题描述运动过程的认识深度.评述介绍了三种建模思路:刚体平面运动模型,最优控制模型和微分方程模型,其中微分方程模型是在阅卷过程中从考生答卷中发现的,从答卷中发现了不少有共性的毛病,反应出国内大学和研究生数学教育中的弊病,值得有识之士们重视.