在有限差分法下一类抛物型方程的离散化

本文对于一类带有狄拉克函数δ０初值的抛物型方程，在有限差分法下进行离散化，证明了其数值解的存在性、唯一性，尤其是它的稳定性．     

抛物型积分微分方程的非协调$H^{1}$-Galerkin混合有限元方法

H^1-Galerkin nonconforming mixed finite element methods are analyzed for integro-differential equation of parabolic type. By use of the typical characteristic of the elements, we obtain that the Galerkin mixed approximations have the same rates of convergence as in the classical mixed method, but without LBB stability condition.     

Inverse Problem for an Integro-Differential Equation of Acoustics

We consider the hyperbolic integro-differential equation of acoustics. The direct problem is to determine the acoustic pressure created by a concentrated excitation source located at the boundary of a spatial domain from the initial boundary-value problem for this equation. For this direct problem, we study the inverse problem, which consists in determining the onedimensional kernel of the integral term from the known solution of the direct problem at the point x = 0 for t > 0. This problem reduces to solving a system of integral equations in unknown functions. The latter is solved by using the principle of contraction mapping in the space of continuous functions. The local unique solvability of the posed problem is proved.     

非光滑时间分数阶齐次扩散方程的修正

当初值不光滑时,时间分数阶齐次扩散方程数值方法的精度会下降.为了得到高阶时间收敛格式,提出加权移位的Griinwald-Letnikov的修正格式,运用Lubich的修正方法,得到非光滑时间分数阶齐次扩散方程的收敛阶仍为O(k2).最后,通过数值算例验证了数值计算结果与理论计算结果一致.     

带扩散项四阶抛物方程的一类本性并行差分格式

给出逼近带扩散项四阶抛物方程一组非对称差分格式,对此组非对称格式重新组合,得到了一类新的具有并行本性的算法.随后,利用矩阵法证明了算法的绝对稳定性.最后给出数值实验.     

带时滞的退化非线性抛物方程的熄灭

该文研究一带时滞的退化非线性抛物方程的初边值问题。运用正则化方法和上下解技巧证明了上述问题的古典正解的局部存在性及其可延拓性。讨论了整体存在性和 有限时刻熄灭，建立了临界长度，得到了熄灭点的位置以及特殊f(u)情形下的熄灭速率估计。     

高维抛物型方程差分格式的稳定性

本文提出了一个改进抛物型方程差分格式稳定性条件的新方法,给出并证明新方法稳定的充要条件,数值例子显示了本方法的计算优越性.     

On a Parabolic Type Fredholm Integrodifferential Equation

In this paper we study some basic properties of solutions of a parabolic-type Fredholm integrodifferential equation. The Banach fixed point theorem and the variant of a certain integral inequality with explicit estimate are used to establish the results.     

双调和型抛物方程的Schauder估计

该文给出了双调和型抛物方程初值问题解的Schauder估计,并且在适当的空间中证明了解的存在性与惟一性．类似于二阶情形,首先通过Fourier变换得到一个形式解,然后再利用位势理论和逼近方法得到解的正则性、唯一性及存在性．该方法简单而易懂．     

On an Inverse Problem for a Parabolic Equation

In this paper, we study the inverse problem of the reconstruction of the right-hand side of special form for a parabolic equation in u in which the coefficients of u _t and u depend on u _(x,t) , with overdetermination given by the integral of the solution over time. The Fredholm property for this problem and the existence and uniqueness theorems in Sobolev spaces are established.     

抛物方程时间连续有限元全离散格式的超收敛性

利用张量积分解和时间方向单元正交分解,证明了线性抛物型方程的时间连续全离散有限元在单元节点和内部的特征点的超收敛性.并用连续有限元计算了非线性Schrodinger方程,验证了能量的守恒性.计算结果与理论相吻合.     

The Quantum Faedo-Galerkin Equation: Evolution Operator and Time Discretization

Time dependent quantum systems have become indispensable in science and nanotechnology. Disciplines including chemical physics and electrical engineering have used approximate evolution operators to solve these systems for targeted physical quantities. Here, we discuss the approximation of closed time dependent quantum systems on bounded domains via evolution operators. The work builds upon the use of weak solutions, which includes a framework for the evolution operator based upon dual spaces. We are able to derive the corresponding Faedo-Galerkin equation as well as its time discretization, yielding a fully discrete theory. We obtain corresponding approximation estimates. These estimates make no regularity assumptions on the weak solutions, other than their inherent properties. Of necessity, the estimates are in the dual norm, which is natural for weak solutions. This appears to be a novel aspect of this approach.     

Problems on a Semiaxis for an Integro-Differential Equation with Quadratic Nonlinearity

Computational Mathematics and Mathematical Physics - A functional equation is considered in which a linear combination of a two-variable function and its time derivative is set equal to the double...     

On Discretization Schemes for Stochastic Evolution Equations

Stochastic evolutional equations with monotone operators are considered in Banach spaces. Explicit and implicit numerical schemes are presented. The convergence of the approximations to the solution of the equations is proved. Mathematics Subject Classifications (2000) Primary: 60H15; Secondary: 65M60.István Gyöngy: This paper was written while the first named author was visiting the University of Paris X. The research of this author is partially supported by EU Network HARP.Annie Millet: The research of the second named author is partially supported by the research project BMF2003-01345.     

Numerical Schemes for Rough Parabolic Equations

This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0,1) perturbed by a non-linear rough signal. It is the continuation of Deya (Electron. J. Probab. 16:1489–1518, 2011) and Deya et al. (Probab. Theory Relat. Fields, to appear), where the existence and uniqueness of a solution has been established. The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs. The results apply to a geometric 2-rough path, which covers the case of the multidimensional fractional Brownian motion with Hurst index H>1/3.     

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.     

Periodic Solution of Nonlinear Parabolic Differential Equation With Time Lag

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Asymptotic Weighted-Periodicity of the Impulsive Parabolic Equation with Time Delay

The main aim of this paper is to investigate the effects of the impulse and time delay on a typeof parabolic equations.In view of the characteristics of the equation,a particular iteration scheme is adopted.The results show that Under certain conditions on the coefficients of the equation and the impulse,the solutionoscillates in a particular manner—called"asymptotic weighted-periodicity".