二粒子Boltzmann方程组的奇异扰动解法:初始层解

本文讨论了二粒子Boltzmann方程组的初始层解.为此先对未知变量进行了Fourier变换,然后运用奇异扰动解法得到了二粒子Boltzmann方程组的正规解和初始层解以及其初始层解的初级和高级近似,并且得到了初始层解和正规解的连接.     

一类对偶积分方程组正则化为Cauchy奇异积分方程组解法

本文基于Mellin变换法求解复杂更一般形式的对偶积分方程组．通过积分变换，由实数域化成复数域上的方程组，引入未知函数的积分变换，移动积分路径，应用Cauchy积分定理，实现退耦正则化为Cauchy奇异积分方程组，由此给出一般性解，并严格证明了对偶积分方程组退耦正则化为Cauchy奇异积分方程组与原对偶积分方程组等价性，以及对偶积分方程组解的存在性和唯一性．给出的解法和理论解，作为求解复杂对偶积分方程组一种有效解法，可供求解复杂的数学、物理、力学中的混合边值问题应用．     

Boltzmann方程的两个解的平均作为二粒子Boltzmann方程系的解

在这篇文章,由Boltzmann方程的两个解构造出二粒子Boltzmann方程系的一类解。利用这个结果,求出了二粒子Boltzmann方程系的平衡解及宏观运动方程组。     

一类奇异积分方程组的样条间接近似解法

本文利用三次复插值样条函数给了定义于复平面上光滑封闭曲线上的一类奇异积分方程组（１）的一种间接近似解法，讨论了误差估计和一致收敛性。     

多元多项式齐次方程组的一个解法

本文对多元多项式齐次方程组给出一种解法，并给出其解子模的基所含向量的个数在最坏情况时的下界。     

一类非线性Dirac方程组

本文寻求方程组 i sum from j=0 to 3 r~j _jΨ+L(Ψ■)Ψ=0的某些含两个零分量的解.     

二维色散长波方程组的精确解

利用齐次平衡法给出了二维色散长波方程组的定态解、孤立波解与非孤立波解等几种显式精确解。这个方法也可用来寻找其它非线性发展方程的不同类型的精确解。     

奇异半线性反应扩散方程组Cauchy问题

本文讨论如下问题其中σ>0,Pi>1,qi>1(i=1,2),α1≥0,α2>0,β1>0,β2≥0,fi(x)(i=1,2)连续有界非负, (f1(x),f2(x))(?)(0,0).给出了非负局部解存在的几个充分条件和解的爆破结果.     

范德蒙型方程组最小二乘解的快速算法

在用多项式进行曲线拟合等实际问题中,需要求解以范德蒙型矩阵VT为系数阵的线性方程组VTx=b的最小二乘解.     

二次谐波产生耦合方程组的解

本文讨论了非线性光学中的二次谐波产生的耦合方程组 ,利用变分法证明了耦合方程组非平凡解的存在性 ,然后进行了数值模拟 ,实验结果表明文中方法比经典的非线性光学中的方法有较大的改进 ,这对优化光倍频器件的设计将有所帮助 .     

关于奇异摄动问题一致有效渐近解的构造方法

本文指出文［１］的不足，提出构造在整个区域内一致有效渐近解的方法，解决了长期以来未能解决的问题．     

Existence of nonlinear boundary layer to the Boltzmann equation with reverse reflection boundary condition

Many physical models have boundaries. When the Boltzmann equation is used to study a physical problem with boundary, there usually exists a layer of width of the order of the Knudsen number along the boundary. Hence, the research on the boundary layer problem is important both in mathematics and physics. Based on the previous work, in this paper, we consider the existence of boundary layer solution to the Boltzmann equation for hard sphere model with positive Mach number. The boundary condition is imposed on incoming particles of reverse reflection type, and the solution is assumed to approach to a global Maxwellian in the far field. Similar to the problem with Dirichlet boundary condition studied in [S. Ukai, T. Yang, S.H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 3 (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian. Moreover, there is an implicit solvability condition on the boundary data. According to the solvability condition, the co-dimension of the boundary data related to the number of the positive characteristic speeds is obtained.     

一维奇异p-Laplace方程的上下解方法[英文]

本文讨论了一维奇异 p Laplace方程( φp( y′) )′+ q(t) f(t,y) =0 ,0 相似文献   

The nonlinear boundary layer to the Boltzmann equation with mixed boundary conditions for hard potentials

In this paper, the existence of boundary layer solutions to the Boltzmann equation for hard potential with mixed boundary condition, i.e., a linear combination of Dirichlet boundary condition and diffuse reflection boundary condition at the wall, is considered. The boundary condition is imposed on the incoming particles, and the solution is supposed to approach to a global Maxwellian in the far field. As for the problem with Dirichlet boundary condition (Chen et al., 2004 [5]), the existence of a solution highly depends on the Mach number of the far field Maxwellian. Furthermore, an implicit solvability condition on the boundary data which shows the codimension of the boundary data is related to the number of the positive characteristic speeds is also given.     

Existence of nonlinear boundary layer solution to the Boltzmann equation with physical boundary conditions

In this paper, the existence of boundary layer solutions to the Boltzmann equation with two physical boundary conditions for hard sphere model is considered. The boundary condition is first imposed on incoming particles of diffuse reflection type and the solution tends to a global Maxwellian in the far field. Similar to the problem with Dirichlet boundary condition studied in [S. Ukai, T. Yang, S.H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 236 (3) (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian, and there is an implicit solvability conditions yielding the co-dimensions of the boundary data. At last, the specular reflection boundary condition is considered and the similar conclusions are obtained.     

四阶椭圆型方程奇异摄动问题的数值解

本文对一类四阶椭圆型方程奇异摄动问题建立了指数型拟合差分格式,并且证明了这种格式在能量范数意义下关于小参数ε的一致收敛性.最后,我们给出了数值结果.     

周期单元中相对论Boltzmann方程初值问题整体解的存在性

本文证明了周期单元中带有某些硬相互作用的相对论Ｂｏｌｔｚｍａｎｎ方程初值问题在初值满足质量、能量和熵有限的条件下具有一个整体温和解．     

椭圆型方程的内层和边界层渐近解

本文讨论了一类半线性椭圆型方程边值问题.利用微分不等式理论,研究了边值问题内层和边界层解的存在性和渐近性态.     

具有多重解的非线性Robin问题的奇摄动[英文]

本文利用边界层法，研究了具有多重解的非线性Robin问题εx″ f(t,x)x′ g(t,x)=0,0≤t≤1,x′(0,ε）-ax(0,ε)=A,x′(1,ε） bx(1,ε）=B其中ε为正的小参数。在适当的假设下，我们通过给出外部解展开式系数的一般表达式，得到了退化问题的边值为某方程的多重根时的渐近解，推广了有关结果。