A fourth-order boundary value problem for a Sturm-Liouville type equation

An existence result of multiple solutions for a fourth-order Sturm-Liouville boundary value problem with variable parameters is established. As a consequence, three solutions for a boundary value problem with a fourth-order equation in a complete form are obtained. Our approach is based on variational methods.     

A criterion for correct solvability of the Sturm-Liouville equation in the space

We consider an equation

where , and By a solution of equation (1), we mean any function such that and equality (1) holds almost everywhere on In this paper, we obtain a criterion for the correct solvability of (1) in ,

     

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.     

Criteria of the three cases for non-self-adjoint singular Sturm-Liouville difference equations

This paper is concerned with the criteria of the three cases for non-self-adjoint singular Sturm-Liouville difference equations. It has been known that this class of equations was classified into cases I, II and III by R.H. Wilson (Proc. R. Soc. A 401 (2005)). In this paper, several criteria of the cases I, II and III are established, respectively.     

Asymptotic expansions for the Sturm-Liouville problem by homotopy perturbation method

The asymptotic formulae for the eigenvalues and eigenfunctions of Sturm-Liouville problem with the Dirichlet boundary conditions when the potential is square integrable on [0, 1] are obtained by using homotopy perturbation method.     

Parametrization of the isospectral set for the vector-valued Sturm-Liouville problem

We obtain a parametrization of the isospectral set of matrix-valued potentials for the vector-valued Sturm-Liouville problem on a finite interval.     

A two-stage difference method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped

A novel two-stage difference method is proposed for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. At the first stage, approximate values of the sum of the pure fourth derivatives of the desired solution are sought on a cubic grid. At the second stage, the system of difference equations approximating the Dirichlet problem is corrected by introducing the quantities determined at the first stage. The difference equations at the first and second stages are formulated using the simplest six-point averaging operator. Under the assumptions that the given boundary values are six times differentiable at the faces of the parallelepiped, those derivatives satisfy the Hölder condition, and the boundary values are continuous at the edges and their second derivatives satisfy a matching condition implied by the Laplace equation, it is proved that the difference solution to the Dirichlet problem converges uniformly as O(h ⁴lnh ^?1), where h is the mesh size.     

Cauchy problem for the multi-dimensional Boussinesq type equation

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for the multi-dimensional Boussinesq type equation utt−Δu+Δ²u=Δσ(u). It proves that the Cauchy problem admits a global weak solution under the assumptions that σ∈C(R), σ(s) is of polynomial growth order, say p (>1), either , s∈R, where β>0 is a constant, or the initial data belong to a potential well. And the weak solution is regularized and the strong solution is unique when the space dimension N=1. In contrast, any weak solution of the Cauchy problem blows up in finite time under certain conditions. And two examples are shown.     

Central difference regularization method for inverse source problem on the Poisson equation

In this paper, we consider an inverse problem of determining an unknown source for the Poisson equation. Since this problem is mildly ill-posed, we apply a central difference regularization method to solve this problem. Furthermore, the convergence estimate is established under a priori choice of the regularization parameter. Some numerical results verify that the proposed method is stable and effective.     

Sturm-LiouVille算子部分谱信息的逆问题

研究了Sturm-Liouville算子Aq,h,Hj,j=1,2,…中势函数q(x)的确定性问题,即已知部分区间[a,1](a∈(0,1))上的势函数q(x),则无限组部分谱信息可唯一确定整个区间[0,1]上的势函数.推广了Simon的方法,且将选择条件的范围从一组谱扩展到了无限组.     

Uniqueness theorems for an impulsive Sturm-Liouville boundary value problem

In this study, an impulsive boundary value problem, generated by Sturm-Liouville differential equation with the eigenvalue parameter contained in one boundary condition is considered. It is shown that the coefficients of the problem are uniquely determined either by the Weyl function or by two given spectra.     

Two positive solutions of a boundary value problem for difference equations

We consider a second order vector boundary value problem for difference equations and establish criteria for the existence of at least two positive solutions by an application of a fixed point theorem in cones.     

Convergent difference schemes for the Hunter-Saxton equation

We propose and analyze several finite difference schemes for the Hunter-Saxton equation

(HS)

This equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to the derivation of several a priori estimates that yield some basic convergence results, is to prove strong convergence of the discrete spatial derivative of the numerical approximations of , which is achieved by analyzing various renormalizations (in the sense of DiPerna and Lions) of the numerical schemes. Finally, we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results.

     

一类分数阶差分方程边值问题多重正解的存在性

考虑一类高阶分数阶差分方程边值问题.构造相关的格林函数,利用不等式技巧,分析格林函数的特征性质.运用不动点指数理论,获得了该分数阶差分方程边值问题存在多重正解的充分条件,举例说明了所获理论的有效性.     

Finite difference methods for half inverse Sturm-Liouville problems

We investigate finite difference solution of the Hochstadt-Lieberman problem for a Sturm-Liouville operator defined on (0, π): given the value of the potential q on (c, π), where 0 < c < π, use eigenvalues to estimate q on (0, c). Our methods use an asymptotic correction technique of Paine, de Hoog and Anderssen, and its extension to Numerov’s method for various boundary conditions. In the classical case c = π/2, Numerov’s method is found to be particularly effective. Since eigenvalue data is scarce in applications, we also examine stability problems associated with the use of the extra information on q when c < π/2, and give some suggestions for further research.     

About an unsolved stability problem for a stochastic difference equation with continuous time

An unsolved problem of stability for stochastic difference equation with continuous time is proposed for consideration.     

On the mixed problem for quasilinear pseudoparabolic equation

In this paper the multidimensional mixed problem for the quasilinear pseudoparabolic equation u_t-L_xu-εL_xu_t=f(t,x,u) is considered. L_x is a differential operator, which composes (with boundary operator) a self adjoint operator. An existence, uniqueness and also continuous dependense on the small parameter ε>0 of generalized solution is proved. The estimation of the difference of exact and approximate solutions is obtained