关于几类奇摄动方程边值问题的渐近与数值分析

硕士论文-《几类奇摄动方程边值问题的渐近与数值分析》

【摘要】 奇异摄动理论及方法是一门非常活跃和不断拓宽的学科.奇异摄动的各种方法已经被广泛应用于自然科学的各个领域,在解决实际问题中显示出较大的功效,大量的动态数学模型都含有小参数,对非线性的复杂方程在无法求出精确解的前提下,求出一致有效的渐近解或数值近似解尤其重要.在实际应用中,数值计算与渐近方法不是相互排斥,而是相互补充的.本文对含有小参数的常微分方程初边值问题解的性质进行了研究,主要研究内容分述如下:1、考虑了一类奇异摄动非线性抛物方程初边值问题,在适当的条件下,首先求出了原问题的外部解,然后利用伸长变量和幂级数展开理论构造出解的高阶形式渐近展开式.2、在适当的条件下研究了一类部分耗散反应扩散系统出现角层现象的奇异摄动问题.首先,构造了退化问题的平衡解,并借助上下解方法来证明所得结果.其次,利用伸长变量,构造了解的初始层项.然后,利用微分不等式理论,研究了初始边值问题解的渐近性态,并讨论了原问题解的存在、唯一性.3、使用有限元法讨论了一类存在单边界层解的奇异摄动对流扩散方程Robin两点边值问题,并借助Shishkin网格法以某种能量范数的形式给出了近似解的误差估计.

【Abstract】 The theory and method for singular perturbation is a very active and constantly broadened subject. All sorts of methods for singular perturbation have been widely applied in many fields of natural science, which play a crucial role in solving practical problems. Most of dynamic mathematical models contain small parameters, so obtaining the uniformly valid asymptotic solution or numerical approximate solution is particularly important for the complex nonlinear equations under the premise of being unable to get the accurate solution. In practical application, the numerical calculation and asymptotic method are not mutually exclusive but complement each other.The properties of the initial-boundary value problem for the ordinary differential equation containing small parameters are studied. The main contents of this paper are outlined as follows:1. The nonlinear initial-boundary value problems for a class of singularly perturbed parabolic equations are considered. Under suitable conditions, firstly, the outer solution of original problem is solved. And then, by using the method of stretched variable and the expanding theory of power series the higher order formal asymptotic expansions of the solutions are constructed.2. A class of singularly perturbed partly dissipative reaction diffusion systems in case of exchange of stabilities are studied under suitable conditions. Firstly, the families of equilibria of the degenerate problems are constructed. The proofs of the results are based on the method of lower and upper solutions. Secondly, by using the stretched variable, the initial layer term of solution is constructed. And then, by using the theory of differential inequalities the asymptotic behaviors of solutions for the initial boundary value problems are studied. Also, the existence and uniqueness of solutions for original problem are discussed.3. A singularly perturbed advection–diffusion two point Robin boundary value problem whose solution has a single boundary layer is studied. Based on finite element method is applied on the problem. Estimation of the error between solution and the finite element approximation are given in energy norm on shishkin-type mesh.

代写硕士论文-【关键词】 奇摄动； 初边值问题； 微分不等式； 渐近展开式； 反应扩散； 对流扩散； 有限元法； Shishkin网格；

【Key words】 Singular perturbation； Initial boundary value problem； Differential inequality； Higher-Order asymptotic expansion； reaction diffusion； advection–diffusion； finite element method； Shishkin mesh