二阶差分方程的奇性边值问题

主要采用上下解方法,研究了一类带有奇性的二阶差分方程的两点边值问题,给出了一系列正解的存在条件.     

A Galerkin method with modified piecewise polynomials for solving a second-order boundary value problem

Summary The classical Ritz-Galerkin method is applied to a linear, second-order, self-adjoint boundary value problem. The coefficient functions of the operator exhibit a piecewise smooth behaviour characteristic of some physical situations. A trial function is constructed using a modified quintic smooth Hermite space , in order to meet some desired regularity conditions for the approximate solution. A collocation technique is used to reduce the amount of computational work. Known convergence properties for the projection method are recalled which, in this particular case, are illustrated by a series of numerical experiments.     

A second-order splitting method for a finite difference scheme for the Sivashinsky equation

A finite difference method is analyzed for the approximation of the solution of the evolutionary fourth-order in space, Sivashinsky equation.We prove that the scheme has a unique solution and we study error estimation for the numerical scheme.     

Positive solution of second-order multi-point boundary value problems for finite difference equation with a p-Laplacian

In this paper, we consider discrete second-order multi-point boundary value problem with a p-Laplacian. By giving condition on f and applying Krasnosel’skii fixed point theorem, we ensure the existence of at least one positive solution and show the existence of eigenvalue intervals.     

非线性渗流耦合系统动边值问题二阶迎风分数步差分方法

对多层非线性渗流方程耦合系统三维动边值问题, 提出适合并行计算的一类二阶迎风分数步差分格式, 利用区域变换、变分形式、能量方法、隐显格式的相互结合、差分算子乘积交换性、高阶差分算子的分解、先验估计的理论和技巧, 得到收敛性的最佳阶l² 误差估计. 该方法已成功地应用到多层油资源运移聚集的资源评估生产实际中, 得到了很好的数值模拟结果.     

A difference scheme for a problem of elasticity theory in spherical coordinates

Translated from Issledovaniya po Prikladnoi Matematike, No. 8, pp. 41–49, 1980.     

Bilateral difference method for solving the boundary value problem for an ordinary differential equation

A method is proposed for calculating the bilateral approximations of the solution of the boundary value problem on [0, 1] for the equation y+p(x)y-q(x)y=f(x) and the derivative of the solution having the maximum deviation O(h² (h)+h³) on {kh} _k ^N =0, where(t) is the sum of the continuity moduli of the functions p, q,f, on the set of points {kh} _k ^N =0, h=1/N by means of O(N) operations. The data obtained for fairly smooth p, q,f allow interpolation to be used for calculating the bilateral approximations of the solution and its higher derivatives having the maximum deviation O(h³) on [0, 1].Translated from Matematicheskie Zametkii, Vol. 11, No. 4, pp. 421–430, April, 1972.     

Isoperimetric estimates of the second boundary value problem for the Poisson equation

Translated from Matematicheskie Zametki, Vol. 50, No. 5, pp. 48–53, November, 1991.     

A finite element method for a singularly perturbed boundary value problem

Summary We examine the problem:u+a(x)u–b(x)u=f(x) for 0<x<1,a(x)>0,b(x)>, ² = 4>0,a, b andf inC ² [0, 1], in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh ², whereC is independent ofh and ). With a natural choice of trial functions, uniform first order accuracy is obtained in theL (0, 1) norm. On choosing piecewise linear trial functions (hat functions), uniform first order accuracy is obtained in theL ¹ (0, 1) norm.     

Inhomogeneous boundary value problem for a second-order singular elliptic equation in a plane sector

We consider an inhomogeneous boundary value problem in a plane sector for a model second-order singular elliptic equation. We prove the well-posed solvability of the problem in weighted function spaces of special form.     

Finite difference methods for a nonlocal boundary value problem for the heat equation

Three different finite difference schemes for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval are considered. The schemes are based on the forward Euler, the backward Euler and the Crank-Nicolson methods. Error estimates are derived in maximum norm. Results from a numerical experiment are presented.     

A singularly perturbed boundary value problem for a second-order equation in the case of variation of stability

A boundary value problem for a second-order nonlinear singularly perturbed differential equation is considered for the case in which there is variation of stability caused by the intersection of roots of the degenerate equation. By the method of differential inequalities, we prove the existence of a solution such that the limit solution is nonsmooth. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 354–362, March, 1998. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00694.     

A fast direct algorithm for implementing a high-order finite element method on rectangles as applied to boundary value problems for the Poisson equation

Fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for a high-order finite element method (FEM) are proposed based on the fast discrete Fourier transform. They generalize logarithmically optimal Fourier algorithms for solving boundary value problems for Poisson-type equations on rectangular meshes to high-order FEM. The algorithms can be extended to the multidimensional case and can be applied to nonstationary problems.     

一类二阶差分方程泛函边值问题的多解性

考虑二阶差分方程泛函边值问题△2u(k-1)=(Fu)(k),k∈[a+1,b-1]z,ω(u)=A,γ(△u)=B多个解的存在性,并获得一个严格单调递增解和一个严格单调递减解.其中a,b∈Z,满足b≥a+2,F为连续算子,ω,γ均为连续泛函.     

一类四阶差分方程多点边值问题的正解

为解决多点支撑弹性梁的正解的存在性问题,运用锥上不动点指数理论,研究一类含参四阶差分方程多点边值问题.获得了当参数在一定范围内取值时正解的存在性结果,得到了正解存在的充分条件.