三维广义神经传播型非线性拟双曲方程(组)的整体强解

<正> 1962年,Nagumo等提出了神经传播方程u_(tt)=u_(xxt)-c_1(1-u+c_2u~2)u_t-u,(1.1)c_1,c_2为非负常数.1963年,Arima,Yamaguti等把方程(1.1)推广为u_(tt)=u_(xxt)-f(u)u_t-g(u).(1.2)1975年,Pao,C.V研究了下述更为广泛的多维非线性拟双曲方程的初边值问题     

无界可测系数的二阶非线性椭圆型方程解的表示定理与混合边值问题

<正> 在L.Bers和L.Nirenberg的文[1]中,研究了一定条件下的二阶非线性一致椭圆型方程 Φ(x,y,u,u_x,u_y,u_(xx),u_(xy),u_(yy))=0(1.1)于单连通区域上的Dirichlet边值问题与Neumann边值问题解的存在性.近几年来,我们也曾对二阶非线性一致椭圓型方程的复形式讨论过解的一些性质与平面多连通区域D上的第一、二、三边值问题与混合边值问题的可     

变系数KdV-Burgers-Kuramoto方程的群论分析——容许变换及对称群

提出变系数KdV-Burgers-Kuramoto(VCKdVBK)方程$u_t+f(x,t)uu_x+g(x,t)u_{xx}+h(x,t)u_{xxx}+k(x,t)u_{xxxx}=0$的容许变换, 它不改变方程的形式但可能改变方程的系数$f, g, h$及$k$.然后, 基于这个容许变换, 给出了VCKdVBK方程的7种不同类型的等价类. 最后, 也给出了VCKdVBK方程的Lie点对称群.     

具有间断系数的两点边值问题的配置方法

§1 引言 C.de.Boor和B.Swartz讨论过用配置法求边值问题 D~mu=F(x,u,…,D~(m-1)u),x∈I=(0,1), (1.1) β_iu=ci,i=1,2,…,m (1.2)的近似解u_△。当(1.1)式右端充分光滑时,[1]证明了u_△存在唯一,并得到了总体误差和结点误差估计式。     

非线性梁振动方程的周期解

一、引言考虑下述问题Ku″ A~2u M(‖A~1/2u‖~2)Au Au′=f(x,t),t>0,x∈Ω,(1.1)u|_t=0~=u_0(x),x∈Ω,(1.2)Ku′|_(t=0)=u_1(x),x∈Ω,(1.3)u=0,x∈(?)Ω,t≥0 (1.4)的ω-周期解的存在性.其中 Ω(?)R~n 为一有界光滑区域,u′=((?)u)/((?)t),u_″=((?)u)/((?)t)~2,K 为有界线性对称算子且满足(Ku,u)≥0,M∈C~1[0,∞),M(ξ)≥-β,ξ≥0.此模型最初由Woinowsky 和 Krieger 提出,方程形式为     

GLOBAL EXISTENCE OF SOLUTIONS TO INITIAL BOUNDARY VALUE PROBLEM FOR NONLINEAR PARABOLIC EQUATIONS

§1. Introduction In this paper we are concerned with the following initial boundary value problem for nonlinear parabolic equations: where Γ is the smooth boundary of a bounded domain Ω∈R~n, x=(x_1,…x_n), D_xu=(u_(x_1),… u_(x_n)), D_x~2=(u_(x_ix_j)), (i, j=1, …,n). What we want do do in this paper is to consider the global existence of smooth solutions for (1.1) under the assumption of small initial data.     

双曲型方程组高分辨率格式的一种线性场修正方法

§1.引言 本文研究双曲型方程: (?u)/(?t)+(?f(u))/(?x)=0,t>0,-∞相似文献   

寻找最优控制的具有约束算子的梯度算法

§1.具有约束算子的梯度算法 考虑连续的受控系统,其运动轨道及控制满足常微分方程组 x=f(x,u,t),x(t_0)=x_0, (1.1)其中x=(x_1,x_2,…,x_n)~T?E~n是系统的状态变量,u=(u_1,u_2,…,u_r)~T?E~r是系统的控制变量;f(·,·,·)=(f_1(·,·,·),…,f_n(·,·,·))~T是由E~n×E~r×E~1到E~n中的向量值函数;t_0是运动的起始时刻;x_0是运动的初始状态;t_f是运动的终结时刻.为简单起见,下面假定t_0,x_0,t_f均已给定。 我们把定义在区间[t_0,t_f]上的每一个在E~r中取值的分段连续函数u(t)=(u_1(t),u_2(t),…,u_r(t))~T称作系统(1.1)的一个控制。所有这样的控制的集合记作H,给定系     

Asymptotic Behaviors of the Solutions to Scalar Viscous Conservation Laws on Bounded Interval

Abstract This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value prob-lem for scalar viscous conservations laws u_t+f(u)_x=u_(xx) on[0,1],with the boundary condition u(0,t) =u_,u(1,t)=u_+ and the initial data u(x,0)=u_0(x,0)=u_0(x),where u_≠u_+ and f is a given function satisfyingf'(u)>0 for u under consideration.By means of energy estimates method and under some more regular condi-tions on the initial data,both the global existence and the asymptotic behavior are obtained.When u_u_+, which corresponds to shock waves in inviscid conservation laws, it is established for weak shockwaves,which means that │u_-u_+│is small.Moreover,exponential decay rates are both given.     

Hammerstein方程的多解性及其应用

本文在较一般的锥内证明了一类Hammerstein方程至少存在两个非负凸(凹)函数解u_1(t)、u_2(t),并且这两个解相对于γ(t)是可比较的,即,γu_1(t)≤u_2(t).然后应用于两点边值问题.     

EFFICIENT LINEAR SCHEMES WITH UNCONDITIONAL ENERGY STABILITY FOR THE PHASE FIELD MODEL OF SOLID-STATE DEWETTING PROBLEMS

In this paper, we study linearly first and second order in time, uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the novel "scalar auxiliary variable" (SAV) approach, a new developed efficient and accurate method for a large class of gradient flows. The schemes are based on the first order Euler method and the second order backward differential formulas (BDF2) for time discretization, and finite element methods for space discretization. The proposed schemes are proved to be unconditionally stable and the discrete equations are uniquely solvable for all time steps. Various numerical experiments are presented to validate the stability and accuracy of the proposed schemes.     

关于色散方程的一类二阶恒稳显格式

1　引　　言对于具有高阶空间导数的发展方程 ,其显格式因结构简单 ,易于计算 ,具有明显的计算优越性 ,但已有的绝大多数显格式的稳定性条件都十分苛刻 (见 [6 ] -[1 5] ) ,远不如一般隐格式 ,使其应用受到限制 .1 994年《计算物理》中关于“色散方程的一类具任意稳定性的显格式”一文 (见 [1 4 ] ) ,把色散方程显格式的稳定性条件提高到了可以任意选择的程度 ,但截断误差仅为 O(τ+h) .本文构造了新一类双参数显式差分格式 ,它是绝对稳定的 ,且其截断误差是 O(τ+h2 ) ,它结构简单 ,易于实现计算 ,利于实际应用 .我们用数值例子验证了理论…     

High‐order Padé and singly diagonally Runge‐Kutta schemes for linear ODEs,application to wave propagation problems

In this article, we address the problem of constructing high‐order implicit time schemes for wave equations. We consider two classes of one‐step A‐stable schemes adapted to linear Ordinary Differential Equation (ODE). The first class, which is not dissipative is based on the diagonal Padé approximant of exponential function. For this class, the obtained schemes have the same stability function as Gauss Runge‐Kutta (Gauss RK) schemes. They have the advantage to involve the solution of smaller linear systems at each time step compared to Gauss RK. The second class of schemes are constructed such that they require the inversion of a unique linear system several times at each time step like the Singly Diagonally Runge‐Kutta (SDIRK) schemes. While the first class of schemes is constructed for an arbitrary order of accuracy, the second‐class schemes is given up to order 12. The performance assessment we provide shows a very good level of accuracy for both classes of schemes, and the great interest of considering high‐order time schemes that are faster. The diagonal Padé schemes seem to be more accurate and more robust.     

四阶杆振动方程的两类高精度辛格式

In this paper, we present two classes of symplectic schemes with high order accuracy for solving four-order rod vibration equation utt uxxxx=0 via the third type generating function method. First, the equation of four order rod vibration is written into the canonical Hamilton system; second, overcoming successfully the essential difficult on the calculus of high order variations derivative, we get the semi-discretization with arbitrary order of accuracy in time direction for the PDEs by the third type generating function method. Furthermore the discretization of the related modified equation of original equation is obtained. Finally, arbitrary order accuracy symplectic schemes are obtained. Numerical results are also presented to show the effectiveness of the scheme, high order accuracy and properties of excellent long-time numerical behavior.     

方程=f(x)十字架格式

描述了单个含有单位质量的质点在保守力f(x)作用下一维运动的位移.这个系统的主要特点是能量守恒:     

Efficient Simulation of Wave Propagation with Implicit Finite Difference Schemes

Finite difference method is an important methodology in the approximation of waves. In this paper, we will study two implicit finite difference schemes for the simulation of waves. They are the weighted alternating direction implicit (ADI) scheme and the locally one-dimensional (LOD) scheme. The approximation errors, stability conditions, and dispersion relations for both schemes are investigated. Our analysis shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme. Moreover, the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time. In order to improve computational efficiency, numerical algorithms based on message passing interface (MPI) are implemented. Numerical examples of wave propagation in a three-layer model and a standard complex model are presented. Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.     

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

In this article, we study a new second‐order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for higher‐order‐in‐time temporal discretizations is how to ensure an unconditional energy stability without compromising numerical efficiency or accuracy. We propose a framework for designing a second‐order numerical scheme with unconditional energy stability using the BDF method with constant coefficient stabilizing terms. Based on the unconditional energy stability property that we establish, we derive an stability for the numerical solution and provide an optimal convergence analysis. To deal with the highly nonlinear four‐Laplacian term at each time step, we apply efficient preconditioned steepest descent and preconditioned nonlinear conjugate gradient algorithms to solve the corresponding nonlinear system. Various numerical simulations are presented to demonstrate the stability and efficiency of the proposed schemes and solvers. Comparisons with other second‐order schemes are presented.     

Efficient numerical schemes for the solution of generalized time fractional Burgers type equations

The main aim of this paper is to propose two semi-implicit Fourier pseudospectral schemes for the solution of generalized time fractional Burgers type equations, with an analysis of consistency, stability, and convergence. Under some assumptions, the unconditional stability of the schemes is shown. In implementation of these schemes, the fast Fourier transform (FFT) can be used efficiently to improve the computational cost. Various test problems are included to illustrate the results that we have obtained regarding the proposed schemes. The results of numerical experiments are compared with analytical solutions and other existing methods in the literature to show the efficiency of proposed schemes in both accuracy and CPU time. As numerical solution of fractional stochastic nonlinear partial differential equations driven by Brownian motions are among current related research interests, we report the performance of these schemes on stochastic time fractional Burgers equation as well.     

Two-level finite difference scheme of improved accuracy order for time-dependent problems of mathematical physics

In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements, extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues concerning the computational implementation of schemes having higher approximation orders are discussed. From the general point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed.     

High‐order difference schemes for 2D elliptic and parabolic problems with mixed derivatives

We propose a 9‐point fourth‐order finite difference scheme for 2D elliptic problems with a mixed derivative and variable coefficients. The same approach is extended to derive a class of two‐level high‐order compact schemes with weighted time discretization for solving 2D parabolic problems with a mixed derivative. The schemes are fourth‐order accurate in space and second‐ or lower‐order accurate in time depending on the choice of a weighted average parameter μ. Unconditional stability is proved for 0.5 ≤ μ ≤ 1, and numerical experiments supporting our theoretical analysis and confirming the high‐order accuracy of the schemes are presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 366–378, 2007