TARGET PATTERNS AND SPIRAL WAVES IN REACTION-DIFFUSION SYSTEMS WITH LIMIT CYCLE KINETICSAND WEAK DIFFUSION

1.IntroductionTargetpatternsandspiralwavesarecommonlyobservedincertainmodelsofchemicalandbiologicalsystemssuchastheBelousov-ZhabotinskiireactionandthesocialamoebasDictyosteliumdiscoideium(of.11--4]andthereferencestherein).Thesesystemsaregovernedbyachemicalorbiologicalreactionandspatialdiffusion,i.e.reaction-diffusionequations.Generallyspeaking,atargetpatternisasetofconcentricringsofconstantconcentrationeachmovingoutward.Alonganyradialline,thepatternbehavesasymptoticallylikeaperiodictravellin…     

A class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation

A class of nonlinear initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions and using the theory of differential inequalities the asymptotic solution of the initial boundary value problems is studied.     

Convergence phenomena of \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} $\mathcal{Q}_p$ \end{document}‐elements for convection–diffusion problems

We present a numerical study for singularly perturbed convection–diffusion problems using higher order Galerkin and streamline diffusion finite element method. We are especially interested in convergence and superconvergence properties with respect to different interpolation operators. For this we investigate pointwise interpolation and vertex‐edge‐cell interpolation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection–diffusion problem with characteristic layers

We analyze the superconvergence property of the Galerkin finite element method (FEM) for elliptic convection–diffusion problems with characteristic layers. This method on Shishkin meshes is known to be almost first‐order accurate (up to a logarithmic factor) in the energy norm induced by the bilinear form of the weak formulation, uniformly in the perturbation parameter. In the present paper the method is shown to be almost second‐order superconvergent in this energy norm for the difference between the FEM solution and the bilinear interpolant of the exact solution. This supercloseness property is used to improve the accuracy to almost second order by means of a postprocessing procedure. Numerical experiments confirm these results. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

The Nonlinear Singularly Perturbed Nonlocal Reaction Diffusion Systems

In this paper the singularly perturbed initial boundary value problems for a nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and asymptotic behavior of solutions for the problem are studied.     

一类非线性奇摄动非局部反应扩散系统

Recently the nonlinear singularly perturbed problem has been investigated in theinternational academic circles[1 ,2 ] .Approximation methods have been developed andrefined,including the method of averaging,boundary layer method,matched asymptoticexpanision method and multiple scales method.Many scholars such as O' Malley,Jr.[3] ,Butuzov,Nefedov and Schneider[4] ,Kelley[5] ,Mizoguchi,Yanagida and Life[6] have done agreat deal of work.Using the method of differential inequality and other met…     

A class of singularly perturbed reaction diffusion integral differential system

Considertheinitialboundaryvalueproblemforthereactiondiffusionintegraldifferentialsystemoftheformeisapositiveparameter,x~(xl,x29...5x.)Cn,fidenotesaboundedregioninR",offsignifiesthesmoothboundaryoffiandkistheoutwardnormalderivativeonan,j:oofthenolilocaltermsTimisthelineintegralfromxotox.Theauthorhasdiscussedthisproblemundervarioussituationsusingthe.methodofdifferentialineqllalities['--8].Thispaperinvolvesaclassofinitialboundaryvalueproblemsforwidersingularlyperturbednonlinearreactiondiffusioni…     

一类四阶非线性奇摄动方程的边值问题

利用匹配渐近展开法,讨论了一类四阶非线性方程的具有两个边界层的奇摄动边值问题．引进伸长变量,根据边界条件与匹配原则,在一定的可解性条件下,给出了外部解和左右边界层附近的内层解,得到了该问题的二阶渐近解,并举例说明了这类非线性问题渐近解的存在性．     

一类反应扩散方程奇摄动Robin问题的广义解

讨论了一类奇摄动反应扩散方程R ob in问题.在适当的条件下,研究了问题广义解的渐近性态.     

Asymptotic solution of nonlinear nonlocal singularly perturbed reaction diffusion problems with two parameters

A class of differential-difference reaction diffusion equations initial boundary problem with a small time delay is considered. Under suitable conditions and by using method of the stretched variable, the formal asymptotic solution is constructed. And then, by using the theory of differential inequalities the uniformly validity of solution is proved.     

一类具非线性边值条件的非线性方程的奇摄动问题

研究了一类具非线性边值条件的非线性方程的奇摄动问题,运用合成展开法构造了问题的形式渐近解,并用微分不等式理论证明了所得渐近解的一致有效性.     

伴有边界摄动二阶非线性系统的奇摄动

讨论了伴有边界摄动的含积分算子的二阶非线性微分方程组边值问题的奇摄动.在适当的假设条件下,通过对角化技巧,证明了解的存在,并估计了余项.     

Periodic traveling waves and locating oscillating patterns in multidimensional domains

We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in , when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.

     

PSE在边界层中预测转捩位置的应用

层流到湍流的转捩是自然界和各项工程实践中广泛存在的现象,层流和湍流的性质大不相同．因此,预测转捩位置是流体力学中的重要理论和实际问题．针对不可压缩边界层,入口加入展向等幅值型和展向波包型两类扰动,展向等幅值型扰动是由一个二维波(2-D)和两个三维波(3-D)组成,使用抛物化稳定性方程(PSE)的方法来研究扰动的演化和预测转捩位置,并且与数值模拟的结果相比较．结果表明,PSE可以研究扰动的演化和预测转捩位置,同时其计算比数值模拟快得多．     

带非线性无穷大边值条件的二阶半线性奇摄动问题

考虑关于带非线性无穷大边界值条件的二阶半线性奇摄动边值问题.基于边界层校正的思想,分别构造了左、右端点邻域的指数型及代数型的边界层校正函数,得到了问题的渐近解;根据微分不等式理论,获得了该问题解的存在性、渐近解的一致有效性以及渐近解的误差估计.还着重探讨了一定的稳定性条件下问题所能允许的边界值的奇异程度问题.通过一个典型的算例,验证了文中理论结果的正确性以及渐近解的高精度性.     

Layers and corner singularities in singularly perturbed elliptic problems

Some recent work and open problems are reviewed concerning the numerical solution of singularly perturbed elliptic boundary value problems whose solutions have boundary layers and corner singularities. AMS subject classification (2000)  65N15, 65N50, 35J25     

The coupling of hyperbolic and elliptic boundary value problems with variable coefficients

We consider a one‐dimensional coupled problem for elliptic second‐order ODEs with natural transmission conditions. In one subinterval, the coefficient ϵ>0 of the second derivative tends to zero. Then the equation becomes there hyperbolic and the natural transmission conditions are not fulfilled anymore. The solution of the degenerate coupled problem with a flux transmission condition is corrected by an internal boundary layer term taking into account the viscosity ϵ. By using singular perturbation techniques, we show that the remainders in our first‐order asymptotic expansion converge to zero uniformly. Our analysis provides an a posteriori correction procedure for the numerical treatment of exterior viscous compressible flow problems with coupled Navier–Stokes/Euler models. Copyright © 2000 John Wiley & Sons, Ltd.     

具有边界摄动的拟线性Robin问题

A class of quasilinear Robin problems with boundary perturbation are considered. Under suitable conditions, using theory of differential inequalities the asymptotic behavior of solution for the boundary value problem is studied.