双曲-抛物偏微分方程奇异摄动问题的差分解法

本文对双曲-抛物偏微分方程奇异摄动问题构造了一个指数型拟合差分格式.我们不仅在方程中加了一个拟合因子,而且在逼近第二个初始条件时也加了拟合因子.我们利用问题的渐近解证明了差分格式关于小参数的一致收敛性.     

时间导数项含小参数的抛物方程的数值解法

本文讨论时间导数项含小参数的抛物方程.我们依Бахволов构造非均匀网格的差分格式,并证明了格式的一阶一致收敛性.给出了数值结果.     

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.     

双曲-双曲奇异摄动问题的指数型拟合差分格式

本文讨论带有关于x的一阶导数项的双曲奇异摄动初边值问题,在较弱的相容性条件下构造了问题的渐近解并证明了解的一致有效性.然后我们对原问题构造一个指数型拟合差分格式并建立了离散能量不等式.最后我们证明差分问题的解一致收敛于原问题的精确解.     

一类反应扩散方程组的保持正性的差分格式

本文研究一类具有正解的反应扩散方程组的有限差分解法.构造了一个保持正性的差分格式.利用离散的最大值原理证明了差分格式解的非负性,有界性及差分格式的无条件稳定性.这些估计的证明不依赖于微分方程的解而仅仅与初边值条件有关.当微分方程的解适当光滑时,证明了差分格式的一致收敛性.最后给出了数值计算结果,并与以往方法进行了比较.计算结果说明了本文给出的方法的有效性.     

自共轭椭圆型偏微分方程奇异摄动问题的一致收敛差分解法

本文在矩形域内考虑高阶导数项含有小参数的自共轭椭圆型第一边值问题. 本文,我们应用渐近分析方法建立了一种新的差分格式,比较了差分方程的解与微分方程的解的渐近性态,并证明了解的一致收敛性.     

奇异摄动问题的加权差分方法

本文对一阶导数引进了新的差分逼近,给出了一族一致收敛的差分格式.     

奇异摄动半线性抛物方程的数值解法

本文我们讨论了一维和二维奇异摄动半线性抛物方程,我们利用直线法和精确差分格式在非均匀网中上得到了数值解.而且还证明了关于ε的一阶一致收敛性.     

具有初始跳跃的双曲型方程奇异摄动问题的数值解法

本文考虑具有初始跳跃的二阶双曲型方程初边值问题.首先给出解的导数估计.然后在一非均匀网格上构造了一个差分格式,最后在能量范数意义下证明了差分格式解的一致收敛性.     

半线性抛物型方程奇异摄动问题的数值解

本文讨论具有抛物边界层的半线性抛物型方程奇异摄动问题的数值解法,在非均匀网格上构造了两层非线性差分格式,证明了差分格式是一致收敛的,给出了一些数值例子.     

一阶中立型时滞差分方程的线性化振动性定理

本文考虑一类非线性中立型时滞差分方程,证明了在一定条件下,若与此方程相关的一个线性差分方程的每个解振动,则该非线性差分方程的每个解也振动。     

A Stability Theorem for Constrained Optimal Control Problems

This paper presents the stability of difference approximations of an optimal control problem for a quasilinear parabolic equation with controls in the coefficients, boundary conditions and additional restrictions. The optimal control problem has been convered to one of the optimization problem using a penalty function technique. The difference approximations problem for the considered problem is obtained. The estimations of stability of the solution of difference approximations problem are proved. The stability estimation of the solution of difference approximations problem by the controls is obtained.     

非线性Kundu方程的弱守恒差分格式

本文考察了一类非线性Kundu方程的周期初值问题,提出了一种弱守恒的差分格式,对其差分解作了先验估计,证明了格式的收敛性与稳定性,最后,通过数值计算检验了格式的可信性.     

对流扩散方程的本质非振荡特征差分方法

本文把特征差分法[1]和本质非振荡插值[3]相结合,提出了对流扩散方程的本质非荡性征差分格式,避免了基于Lagrange插值特征差分格式在求解解具有大梯度问题时所产生的非物理振荡,并给出了格式的严格误差估计及数值算例。     

Zeros of Difference Polynomials of Meromorphic Functions

This research is a continuation of a recent paper, due to Liu and Laine, dealing with difference polynomials of entire function. In this paper, we investigate the value distribution of difference polynomials of meromorphic functions and prove some difference analogues to some classical results for differential polynomials.     

非线性Gerdjikov-Ivanov方程的守恒差分格式

本文考察了一类非线性Gerdjikov-Ivanov方程的周期初值问题，提出了一种守恒的差分格式，对其差分解作了先验估计．证明了格式的收敛性与稳定性，最后，通过数值计算检验了格式的可信性。     

Construction of Canonical Difference Schemes for Hamiltonian Formalism via Generating Functions

This paper discusses the relationship between canonical maps and generating functions and gives the general Hamilton-Jacobi theory for time-independent Hamiltonian systems. Based on this theory, the general method — the generating function method — of the construction of difference schemes for Hamiltonian systems is considered. The transition of such difference schemes from one time-step to the next is canonical. So they are called the canonical difference schemes. The well known Euler centered scheme is a canonical difference scheme. Its higher order canonical generalisations and other families of canonical difference schemes are given. The construction method proposed in the paper is also applicable to time-dependent Hamiltonian systems.     

Perturbations for linear difference equations

In this paper we study boundary value problems for perturbed second-order linear difference equations with a small parameter. The reduced problem obtained when the parameter is equal to zero is a first-order linear difference equation. The solution is represented as a convergent series in the small parameter, whose coefficients are given by means of solutions of the reduced problem.     

Difference operators,Green's matrix,sampling theory and applications in signal processing

This paper introduces sampling representations for discrete signals arising from self adjoint difference operators with mixed boundary conditions. The theory of linear operators on finite-dimensional inner product spaces is employed to study the second-order difference operators. We give necessary and sufficient conditions that make the operators self adjoint. The equivalence between the difference operator and a Hermitian Green's matrix is established. Sampling theorems are derived for discrete transforms associated with the difference operator. The results are exhibited via illustrative examples, involving sampling representations for the discrete Hartley transform. Families of discrete fractional Fourier-type transforms are introduced with an application to image encryption.     

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.