Alternating Direction Implicit Galerkin Finite Element Method for the Two-Dimensional Time Fractional Evolution Equation

New numerical techniques are presented for the solution of the two-dimensional time fractional evolution equation in the unit square. In these methods, Galerkin finite element is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) method based on the backward Euler method combined with the first order convolution quadrature approximating the integral term are considered. The ADI Galerkin finite element method is proved to be convergent in time and in the $L^2$ norm in space. The convergence order is$\mathcal{O}$($k$|ln $k$|+$h^r$), where $k$ is the temporal grid size and $h$ is spatial grid size in the $x$ and $y$ directions, respectively. Numerical results are presented to support our theoretical analysis.     

Asymptotic Properties for a Semilinear Edge-Degenerate Parabolic Equation

The author deals with a semi-linear edge-degenerate parabolic equation, and proves that the solution increases exponentially under the initial energy J(u₀) ≤ d, where d is the mountain-pass level. Moreover, the author estimates the blow-up time and the blow-up rate for the solution under J(u₀) < 0.     

Long Time Asymptotic Behavior of Solution of Difference Scheme for a Semilinear Parabolic Equation

In this paper we prove that the solution of implicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the corresponding nonlinear stationary problem as $t\rightarrow\infty$. For the discrete solution of nonlinear parabolic problem, we get its long time asymptotic behavior which is similar to that of the continuous solution. For simplicity, we consider one-dimensional problem.     

A High Accuracy Alternating Direction Method for the Wave Equation

The alternating direction implicit (ADI) method of Lees forsolving the wave equation in two space dimensions is generalizedand an ADI method of increased accuracy obtained. The new methodis demonstrated by a numerical example. An extension is alsogiven to cover the case of the wave equation in three spacedimensions.     

Rothe's Method for Semilinear Parabolic Problems with Degeneration

The paper deals with semilinear parabolic initial-boundary value problems whereat the coefficient g(x, t) of the time derivative may vanish at a set of zero measure. Existence of a local weak solution of the problem is proved by means of semidiscretization in time. In order to omit a growth limitation for the nonlinearity we derive uniform boundedness of the approximates in L_∞ (Q_T). Moreover, the weak solution turns out to be continuous even in the points of degeneration.     

Traveling Waves with Multiple and Nonconvex Fronts for a Bistable Semilinear Parabolic Equation

We construct new examples of traveling wave solutions to the bistable and balanced semilinear parabolic equation in \input amssym ${\Bbb R}^N+1$ , $N\geq 2$ . Our first example is that of a traveling wave solution with two non planar fronts that move with the same speed. Our second example is a traveling wave solution with a nonconvex moving front. To our knowledge no existence results of traveling fronts with these type of geometric characteristics have been previously known. Our approach explores a connection between solutions of the semilinear parabolic PDE and eternal solutions to the mean curvature flow in \input amssym ${\Bbb R}^N+1$ .     

Error Estimate of the Fourier Collocation Method for the Benjamin-Ono Equation

In this paper, the Fourier collocation method for solving the generalized Benjamin-Ono equation with periodic boundary conditions is analyzed. Stability of the semi-discrete scheme is proved and error estimate in H1/2-norm is obtained.     

Global Existence and Blow-Up Results for a Classical Semilinear Parabolic Equation

The author studies the boundary value problem of the classical semilinear parabolic equations ut-△u = |u|p-1u inΩ×(0, T), and u = 0 on the boundary × [0, T) and u = φ at t = 0, where Rnis a compact C1domain, 1 < p ≤ p S is a fixed constant, and φ∈ C1 0(Ω) is a given smooth function. Introducing a new idea, it is shown that there are two sets W and Z, such that for φ∈ W, there is a global positive solution u(t) ∈ W with H1omega limit 0 and for φ∈ Z, the solution blows up at finite time.     

Galerkin Approximations for the Linear Parabolic Equation with the Third Boundary Condition

We solve a linear parabolic equation in ^d , d 1, with the third nonhomogeneous boundary condition using the finite element method for discretization in space, and the -method for discretization in time. The convergence of both, the semidiscrete approximations and the fully discretized ones, is analysed. The proofs are based on a generalization of the idea of the elliptic projection. The rate of convergence is derived also for variable time step-sizes.     

On Error Estimate of the Boundary Element Method for Parabolic Equations in a Time-Dependent Interval

This paper discusses the direct boundary element method for parabolic equations in a time-dependent interval. An optimal estimate of the error in maximum norm for the boundary element collocation scheme is given.     

定常的Navier-Stokes方程的非线性Galerkin混合元法及其后验估计

提出了定常的Navier-Stokes方程的一种非线性Galerkin混合元法,并导出非线性Galerkin混合元解的存在性和误差估计及其后验误差估计.     

带扩散项四阶抛物方程一类新的交替分组显格式

首先给出逼近带扩散项四阶抛物方程初边值问题一类非对称差分格式,利用该组非对称格式构造了一类新的交替分组显格式算法,并给出了截断误差分析和绝对稳定性结论,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。     

具时滞的半线性抛物问题

基于C.C.Travis和G.F.Webb等人对时滞问题的研究,作者曾在[5]中对一类被称为带正时滞臂的时滞抛物型问题进行了讨论。本文在[5]的基础上继续讨论此类问题。对于抽象时滞问题,在非线性项满足局部Lip连续(甚至对某些变元仅仅是H?lder连续)及一定的增长性限制的条件下,我们对非线性项作了较精细的估计,用半群方法得到了解的存在性和唯一性。进一步,对带零时滞臂的抛物型问题,我们先用上面的结果建立逼近方程,用紧性方法在较弱的条件下也得到了解的存在性。 其次,我们将所得到的抽象结果应用于二阶时滞抛物型     

一类有界区域上半线性抛物型方程爆破的充分条件

讨论一类基本的半线性抛物型方程,其在物理上对应具有内部热源的热传导问题,提出了一些爆破的充分条件,讨论了有限爆破点与径向对称情况下的爆破点,并证明爆破速率之上界与下界.     

Dependence of the Blow-up Time with Respect to Parameters for Semilinear Parabolic Equations with Nonlinear Memory

In this paper we discuss the bounds for the modulus of continuity of the blow-up time with respect to three parameters of λ, h, and p respectively for the initial boundary value problem of the semilinear parabolic equation.     

黎曼流形上一类非线性抛物方程的Hamilton 梯度估计

本文我们得到了黎曼流形上一类非线性抛物方程的局部Hamilton梯度估计. 利用这个局部估计,我们得到了一个Harnack型不等式和一个Liouville型定理.     

Quenching Time for a Semilinear Heat Equation with a Nonlinear Neumann Boundary Condition

In this paper we consider the finite time quenching behavior of solutions to a semilinear heat equation with a nonlinear Neumann boundary condition. Firstly, we establish conditions on nonlinear source and boundary to guarantee that the solution doesn＇t quench for all time. Secondly, we give sufficient conditions on data such that the solution quenches in finite time, and derive an upper bound of quenching time. Thirdly, undermore restrictive conditions, we obtain a lower bound of quenching time. Finally, we give the exact bounds of quenching time of a special example.