自共轭椭圆型方程的样条解法

本文从二元样条空间的理论出发,构造了一类新的差分格式,并利用它得到了一类自共轭椭圆型方程的样条解,并证明了这样的解的唯一性和收敛性问题.最后,给出了一个数值例子,说明了本方法是可行的.     

一类高阶椭圆型方程Dirichlet问题

本文讨论了一类奇摄动高阶椭圆型方程Dirichlet问题，利用伸长变量和变界层校正法，得到了问题解的形式渐近展开式．再用微分不等式理论，证明了解的一致有效性．     

一类高阶非线性椭圆型方程的比较定理

本文利用极值原理得到了一类2m阶非线性椭圆型微分方程的比较定理。     

一类高阶拟线性椭圆型方程奇摄动边值问题

本文讨论了一类拟线性椭圆型方程奇摄动Dirichlet边值问题．在适当的条件下，利用不动点定理，研究了边值问题解的存在唯一性及其渐近性态．     

非一致椭圆型方程的广义Green函数

该文我们证明了一类齐次线性非一致椭圆型方程广义Ｇｒｅｅｎ函数的存在性和唯一性，并得到了广义Ｇｒｅｅｎ函数的一性性质，推广和发展了Ｇｒｕｔｅｒ，Ｍ．和Ｗｉｄｍａｎ，ｋ．ｏ［２］的相应结果，而且我们证明广义Ｇｒｅｅｎ函数唯一性的方法与［２］中的方法完全不同．     

一类高阶椭圆型方程奇摄动边值问题

本文讨论了一类２ｍ阶奇摄动椭圆型方程边值问题的渐近性态，得到了问题解的一致有效的渐近展开式。     

非一致椭圆型方程的广义Green函数

广义Grccn函数的存在性和唯一性。     

椭圆型方程九点格式的高阶PCG解法

§1.引言 近十余年来预处理共扼梯度法(preconditioned conjugate gradient,简称PCG)有了很大的发展。但在预处理方法方面除块预处理技术以外,主要是使用针对五点格式的ICCG方法。至于七点格式和九点格式的预处理方法,尚未见到有关的讨论。     

具奇异边界条件的一类拟线性椭圆型方程

得到了一类拟线性椭圆型方程爆破解的存在性。     

一类半线性椭圆型方程正整体解的存在性

本文研究了一个半线性椭圆型方程正整体解的存在性问题.利用上下解方法克服了非齐次项出现带来的困难,获得了正整体解存在的必要条件,具体地给出了在此条件下正整体解所处的领域.     

非线性二维Volterra积分方程的一个高阶数值格式

对非线性二维Volterra积分方程构造了一个高阶数值格式.block-byblock方法对积分方程来说是一个非常常见的方法,借助经典block-by-block方法的思想,构造了一个所谓的修正block-by-block方法.该方法的优点在于除u(x_1,y),u(x_2,y),u(x,y_1)和u(x,y_2)外,其余的未知量不需要耦合求解,且保存了block-by-block方法好的收敛性.并对此格式的收敛性进行了严格的分析,证明了数值解逼近精确解的阶数是4阶。     

Philos-type oscillation theorems for second order damped elliptic equations

Using general means, we establish several new Philos-type oscillation theorems for the second order damped elliptic differential equation
Σi,j=1 N Di[aij（x）Djy]＋Σi=1 N bi（x）Diy＋c（x）f（y）=0
under quite general assumptions. The obtained results are extensions of the well-known oscillation results due to Kamenev, Philos, Yan for second order linear ordinary differential equations and improve recent results of Xu, Jia and Ma.     

High order convergent modified nodal bi-cubic spline collocation method for elliptic partial differential equation

A high order modified nodal bi-cubic spline collocation method is proposed for numerical solution of second-order elliptic partial differential equation subject to Dirichlet boundary conditions. The approximation is defined on a square mesh stencil using nine grid points. The solution of the method exists and is unique. Convergence analysis has been presented. Moreover, the superconvergent phenomena can be seen in proposed one step method. The numerical results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.     

Oscillation of second order semilinear elliptic equations with damping

We obtain some new Kamenev-type oscillation theorems for the second order semilinear elliptic differential equation with damping N ∑i,j=1Di[aij（x）Djy]＋N∑i=1bi（x）Diy＋c（x）f（y）=0 under quite general assumptions. These results are extensions of the recent results of Sun [Sun, Y. G.： New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping. J. Math. Anal. Appl., 291, 341-351 （2004）] in a natural way. In particular, we do not impose any additional conditions on the damped functions bi （x） except the continuity. Several examples are given to illustrate the main results.     

On the comparison principle for unbounded solutions of elliptic equations with first order terms

We prove a comparison principle for unbounded weak sub/super solutions of the equation

$\lambda u?\text{div}(A(x)Du)=H(x,Du)\text{ in }\mathrm{\Omega }$

where $A(x)$ is a bounded coercive matrix with measurable ingredients, $\lambda \ge 0$ and $\xi ?H(x,\xi )$ has a super linear growth and is convex at infinity. We improve earlier results where the convexity of $H(x,?)$ was required to hold globally.     

The mixed problem for degenerate elliptic equations of second order

The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the existence of solutions of the above problem for elliptic equations by the above estimates and the method of parameter extension. We use the complex method, namely first discuss the corresponding problem for degenerate elliptic complex equations of first order, afterwards discuss the above problem for degenerate elliptic equations of second order.