Finite difference reaction-diffusion systems with coupled boundary conditions and time delays

This paper is concerned with finite difference solutions of a coupled system of reaction-diffusion equations with nonlinear boundary conditions and time delays. The system is coupled through the reaction functions as well as the boundary conditions, and the time delays may appear in both the reaction functions and the boundary functions. The reaction-diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference equations for both the time-dependent problem and its corresponding steady-state problem. This investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions, monotone convergence of the time-dependent solution to a maximal or a minimal steady-state solution for quasimonotone functions, and local and global attractors of the time-dependent system, including the convergence of the time-dependent solution to a unique steady-state solution. Also discussed are some computational algorithms for numerical solutions of the steady-state problem when the reaction function and the boundary function are quasimonotone. All the results for the coupled reaction-diffusion equations are directly applicable to systems of parabolic-ordinary equations and to reaction-diffusion systems without time delays.     

具有吸引项的反应扩散型和拟抛物方程解的全局死核问题

在文[1、2]文[3、6]基础上,应用文[7、8]的方法和相应结论,在古典解存在且唯一的条件下,结合文[10、11]研究具吸引项的反应扩散方程(1)的死核问题,且进一步讨论了跟一般的非线性抛物型方程(2)解的全局死核问题,得到新的结果和时间估计.     

Global well-posedness of coupled parabolic systems

The initial boundary value problem of a class of reaction-diffusion systems(coupled parabolic systems)with nonlinear coupled source terms is considered in order to classify the initial data for the global existence,finite time blowup and long time decay of the solution.The whole study is conducted by considering three cases according to initial energy:the low initial energy case,critical initial energy case and high initial energy case.For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence,long time decay and finite time blowup are given to show a sharp-like condition.In addition,for the high initial energy case the possibility of both global existence and finite time blowup is proved first,and then some sufficient initial conditions of finite time blowup and global existence are obtained,respectively.     

QUENCHING PROBLEMS OF DEGENERATE FUNCTIONAL REACTION-DIFFUSION EQUATION

This paper is concerned with the quenching problem of a degenerate functional reaction-diffusion equation. The quenching problem and global existence of solution for the reaction-diffusion equation are derived and, some results of the positive steady state solutions for functional elliptic boundary value are also presented.     

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.     

Time-Delayed finite difference reaction-diffusion systems with nonquasimonotone functions

Summary This paper is concerned with finite difference solutions of a system of reaction-diffusion equations with coupled nonlinear boundary conditions and time delays. The reaction functions and the boundary functions are not necessarily quasimonotone, and the time delays may appear in the reaction functions as well as in the boundary functions. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. Some monotone iteration processes for the finite difference systems are given, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solution is discussed. The asymptotic behavior result leads to some local and global attractors of the time-dependent problem, including the convergence of the time-dependent solution to a unique steady-state solution. An application and some numerical results to an enzyme-substrate reaction-diffusion problem are given. All the results are directly applicable to parabolic-ordinary systems and to reaction-diffusion systems without time delays. The work of this author was supported in part by the National Natural Science Foundation of China No.10571059, E-Institutes of Shanghai Municipal Education Commission No. E03004, Shanghai Priority Academic Discipline, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.     

反应扩散方程的连续时空有限元方法

本文研究了反应扩散方程的连续时空有限元方法.首先建立了其连续时空有限元格式并证明了有限元解的存在唯一性及稳定性.然后通过引入时空投影算子在没有时空网格限制的条件下给出其近似解在节点处的L~2,H~1最优范数估计以及全局L~2(L~2),L~2(H~1)最优范数估计.最后给出两个数值算例来验证方法的有效性与灵活性并说明结论的正确性.     

Numerical Solution of the Reaction-Diffusion Equation

In this paper, we consider the numerical solution for the reaction-diffusion equation. A finite difference scheme and the basic error equality are given. Then the error estimations are proved for the periodic problem with $v(x,t)\geq 0$, the first and second boundary value problems with $v(x,t)\geq v_0＞0$, and for $v(U)\geq v_0›0$. Under some conditions such estimations imply the stabilities and convergences of the schemes.     

带有多项式非线性项的高维反应扩散方程有限差分格式的长时间行为

本文考虑了一类带有多项式非线性项的高维反应扩散方程．建立了一个全离散的有限差分格式,并证明了差分解的存在唯一性．分析了由差分格式生成的离散系统的动力性质,在对差分解先验估计的基础上得到了离散动力系统的整体吸引子的存在性．最后证明了差分格式的长时间稳定性和收敛性．     

Numerical interface curves for some nonlinear diffusion equations

An initial value problem for the generalized Kolmogorov-Petrowsky-Piscunov (nonlinear degenerate reaction-diffusion) equation is studied numerically by the help of a slightly modified finite difference scheme of Douglas-Yanenko-Mimura type. If the initial function has compact support, the solution also will have compact support and an interface appears between the domains where the solution is positive and where it is zero. We present some examples for different parameter values where the numerical solution as well as numerical interfaces behave according to the analytical theory. This revised version was published online in June 2006 with corrections to the Cover Date.     

Long time behavior of solutions for a scalar nonlocal reaction-diffusion equation

In this paper we study the long time behavior of the solution for a scalar nonlocal reaction-diffusion equation, in which the nonlocal term acts to conserve the spatial integral of the power of the unknown function as time evolves. A class of initial data is found to guarantee the existence of positive global solutions and the convergence to some steady states. A sufficient condition for positive global solutions to be unbounded is also given.     

非线性分数阶反应扩散方程组的间断时空有限元方法

利用时间间断空间连续的时空有限元方法构造了空间分数阶反应扩散方程组的可以逐时间层求解的全离散格式.在时间离散区间上,采用Radau积分公式,将插值理论与有限元理论相结合,给出了全离散格式解的存在唯一性结果,并证明了所给格式是无条件稳定的,进而详细给出最优阶L~∞(L~2)模误差估计过程.最后用数值算例验证了理论分析的正确性.     

提高反应—扩散方程有限差分格式的稳定性问题

This paper deals with the special nonlinear reaction-diffusion equation.The finite difference scheme with incremental unknowns approximating to the differential equation (2.1) is set up by means of introducing incremental unknowns methods.Through the stability analyzing for the scheme,it was shown that the stability conditions of the finite difference schemes with the incremental unknowns are greatly improved when compared with the stability conditions of the corresponding classic difference scheme.     

Higher order global solution and normalized flux for singularly perturbed reaction-diffusion problems

In this paper a computational technique is proposed for obtaining a higher order global solution and global normalized flux of singularly perturbed reaction-diffusion two-point boundary-value problems. The HOC (higher order compact) finite difference scheme developed in Gracia et al. (2001) [4] and which is constructed on an appropriate piecewise uniform Shishkin mesh, has been considered to find an almost fourth order convergent solution at mesh points. Using these values, piecewise cubic interpolants based approximations for solution and normalized flux in whole domain have been defined. It has been shown that the global solution and the global normalized flux are also uniformly convergent. Moreover, for the global solution, the order of uniform convergence in the whole domain is optimal, i.e., it is the same as this one obtained at mesh points, whereas, for the global normalized flux, the uniform convergence is almost third order, except at midpoints of the mesh, where it is also almost fourth order. Theoretical error bounds have been provided along with some numerical examples, which corroborate the efficiency of the proposed technique to find good approximations to the global solution and the global normalized flux.     

The reaction-diffusion equation with Lewis function and critical Sobolev exponent

In this paper we consider the existence and asymptotic estimates of global solutions and finite time blowup of reaction-diffusion equation with Lewis function and critical Sobolev exponent.     

LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF A NONLINEAR SCHRODINGER EQUATION WITH WEAKLY DAMPED

1. IntroductionThe nonlinear schr~r equation with weakly dampedwhere t = N, o > 0, together with appropriate boUndary and hatal condition, is ared inmany physical fields. The echtence of an attractor is one of the most boortant ~eristiCSfor a dissipative system. The long-tabs dynamics is completely determined by the attractorof the system. J.M. Ghidaglia[1] studied the lOng-the behavior of the nonlineaz Sequation (1.1) and proved the eAstence of a compact global attractor A in H'(n) which…     

On Persistence and Stability of Some Spatially Inhomogeneous Systems

Local and global stability and persistence of some coupled map lattices (CMLs) and partial differential equations are studied. A logistic CML with noninteger time step and delay is introduced. The persistence results for reaction-diffusion equations are extendable to the telegraph reaction-diffusion equation for a sufficiently small delay parameter. The stability and persistence results are applied to ecology, physics, economics, and immunology.     

A Reaction-diffusion system with nonlinear absorption terms and boundary flux

This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existence and finite time blow-up of the solutions are described via all the six nonlinear exponents appearing in the six nonlinear terms. In addition, we also show the influence of the coefficients of the absorption terms as well as the geometry of the domain to the global existence and finite time blow-up of the solutions for some cases. At last, some numerical results are given.     

Global existence and finite time blow up for a reaction-diffusion system

In this paper, we study the global existence and finite time blow up of positive solution for the initial-boundary value problem of a reaction-diffusion system.     

Spectral problems with mixed Dirichlet–Neumann boundary conditions: Isospectrality and beyond

In this study, an implicit semi-discrete higher order compact (HOC) scheme, with an averaged time discretization, has been presented for the numerical solution of unsteady two-dimensional (2D) Schrödinger equation. The scheme is second order accurate in time and fourth order accurate in space. The results of numerical experiments are presented, and are compared with analytical solutions and well established numerical results of some other finite difference schemes. In all cases, the present scheme produces highly accurate results with much better computational efficiency.