【作者】 马东辉；

【导师】 邹永魁；

【作者基本信息】 吉林大学 ， 计算数学， 2004， 硕士

【摘要】 本文用拟谱方法讨论如下形式的具常迁移率的四阶抛物型Cahn-Hilliard方程的初边值问题：其中u=u（x，t）为相互扩散的两种物质之一的密度，γ＞0为迁移率，假设其为常数。φ（u）≡H’（u），H（u）为双井位势的典型形式： 本文只讨论γ₁=0的情形，此时φ（u）≡H’（u）=-u+γ2u³。 本文的主要目的是要在γ₂＞0时用拟谱方法建立问题（1），（2），（3）的半离散近似并讨论其收敛性。 Cahn-Hilliard方程是一类重要的四阶非线性扩散方程。最初是由Cahn和Hilliard于1958年在研究热力学中两种物质（如合金、聚合物等等）之间相互扩散现象时提出。后来在描述生物种群的竞争与排斥现象，河床迁移过程，固体表面上微滴的扩散等许多扩散现中文摘要东口，研九甲也泥山JI刊件阴鳅罕惧型·从不世纪八十年代以后，人们开始系统的研究cahn一Hilliard方程，近年来由于cahn一Hilliard方程化学，化工和材料科学等方面的重要背景，关于cah。一Hilliard方程的理论研究成果甚多，本文主要从数值方面来研究cahn一Hiiliard方程.所谓拟谱方法是建立在三角插值基础上的配置方法.三角插值具有很好的逼近性质，这使得拟谱方法成为一种高精度的数值方法，选取三角多项式作为逼近空间的基函数，用配方法构找出相应的拟谱近似方程. 本文首先对cahn一Hiiliard方程之初边值问题（l），（2），（3）建立了与之等价的变分形式.证明了如果。:【o，T}、C4{o，二}是方程（z），（2），（3）的解，则。一定是方程 月。，月2。，几2。.02. ，a锐、，a一祝O‘V、、口‘刃、 l下万，刃)十守（下丁二;，丈-下）=（沪（u），二-二二），V刃任C六‘[4、 \月子’一/”\月，2’月，2/、丫\^/’月，2，，，U、岁0火吮)的解.如果。:田，刘。几是方程（4）的解，对任意的t有u（t，.）任e4（o，二）且u满足边界条件（2）及初始条件（3），则它一定是方程（1），（2），（3）的解.然后用拟谱方法对空间变量x作离散近似得到了以下半离散问题.求解方程（a）的拟谱方法为求。N:[0，刘、sN使之满足 几。，几2。为20.几2。.，口肠N、.，口肠N口v、，了\口U\、，_。气，三万，U)N十叭二万于-，石下万)万=气钾Lu万)，又二于少刃，vv七O万， U〔UX“UX妇aX‘。、}，=。=。、。，其中。N。=场。。为。。（x）在靳上的关于离散内积（·，·）N的插值.接下来证明了半离散问题（5），（6）的解的存在唯一性，即证明了当甲>o，吮>0时半离散问题（5），（6）在【0，川上必存在唯一的解.在此基础上证明了半离散问题（s），（e）解的收敛性，即证明了如果。任H;是方 111程（5），（6）的解，则对V，任【O，T」有}{·}{+关‘}，…（·）{{灸“·:CN一+，‘·‘0，，‘，其中c是仅依赖于守，今2和。。（x）的常数.日。（0） 11是初始逼近误差.设u是方程（1），（2），（3）的解，。、是方程（5），（6）的解，我们得到“一“、}{三C!1 00一二N。{1+CN一k.更多还原

【Abstract】 In this paper we consider the initial-boundary value problem of the following forth order parabolic Cahn-Hillard equation with constant rate of transitionwhere u = u(x,t) is the density of one of the two inter diffusion materials, 7 > 0 is the rate of transitio and it is assumed to be constant. (u) = H’(u), H(u) is the standard double-well potentialand in this paper we only consider the case 1 = 0, which gives (u) = -u + u3.The main purpose of this paper is to set up the semi-discrete approximation to the equation (1),(2),(3) with the seudo-spectral method and analyze its convergence property when 2 > 0.Cahn-Hillard equation is a class of important nonlinear diffusion equation with fourth order, which was derived by Cahn and Hillard when they studiedthe phenomenon of inter-diffusion between two materials (such as alloy) in energetic. Later on, such mathematical model is also used by people in describing the phenomenon of the competition and blackball of biology grow groups, the process of riverbed transplant and diffusion of a tiny drop on the surface of a solid. Since 80’ in this century, people have studied the Cahn-Hilliard equation systematically. Recently for the important background of Cahn-Hilliard equation in chemistry, materials, etc, there are great achievement in the research of Cahn-Hillard equation. In this paper we will study Cahn-Hillard equation by the way of numerical method. The seudo-spectral method is a Ritz-Galerkin method, which sets up the semi-discrete scheme for the problem by choosing gonometrical polynomials as the basic functions of the trial function space.We first set up a equivalent variational equation for the initial-boundary value problem of the Cahn-Hilliard equation (1),(2),(3). We prove that if u : [0,T] - C4[0, TT] is a solution of equation (1),(2),(3), u must be a solution of the following equationIf u : [0, T] - HP is a solution of equation (4), and if for any t, u(, t) C4[0, ] and satisfy condition (2),(3), it is a solution of equation (1),(2),(3). We use the seudo-spectral method to deal with the space variable x and get the following semi-discrete problem: find a uN : [0, T] - SN such thatwhere UN0 = INu0 is the interpolation of u0(x) onto the space SN with discrete norm. Then we get the uniqueness and existence property in the whole interval[0,T] for the sime-discrete problem (5),(6). We prove that for 7 > 0, 72 > 0 there is a unique solution in the whole interval for equations (5),(6). Then we obtain the convergence property of the solution of the semi-discrete system (5),(6). We obtain the following estimatewhere e = PNu - UN, u is the solution of (1),(2),(3) and C is a constant depends on 7, 72 and U0(x). We also prove更多还原