摘要
奇异摄动理论及方法是一门应用非常广泛的学科,它是用来求解非线性、高阶或变系数的数学物理方程解析近似解的一种方法,目前的研究非常活跃且在不断拓展。它的主要任务是求含有小参数微分方程的近似解,而这个近似解是通过解一些与原方程有关的较简单方程中得到的,因此被称为解析近似解。用此方法可以对原数学物理问题进行定性或定量的分析和讨论。至今已逐步建立了许多行之有效的奇摄动方法,如匹配渐近展开法,多变量展开法,边界层函数法,V-L方法,PLK方法, WKB方法,补偿紧致方法,多重尺度法,KBM方法,平均变分法,不变域理论和对角化技巧等等。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,在解决实际问题中起到重要的作用,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(近似解)尤其重要。在实际应用中,数值计算与渐近方法都是
求解近似解的有效方法,且相互补充。本文对含有小参数的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、讨论了一类奇摄动反应扩散方程问题,在适当的条件下,首先求出原问题的外部解,然后利用多重尺度法和幂级数渐近展开理论,研究问题广义解的存在,唯一性及其渐近性态。
2、在适当条件下研究具有边界摄动的非线性反应扩散方程的奇摄动Robin问题,并运用微分不等式理论,讨论原问题解的存在性、唯一性及其一致有效的渐近估计。
3、讨论了一类非线性中立型奇异摄动微分差分方程初边值问题,并基于Shishkin网格,运用B-样条配置法和有限差分法对问题进行分析和计算并给出了近似解的误差估计。
The theory and method of application for singular perturbation is a very broad range of subject. The singular perturbation is a method to find approximate analytical solutions of nonlinear, high order, or a mathematical equation with variable coefficients. The current research is very active and constantly expanding .Its main idea is to find approximate solutions with differential equations containing small parameter which is from some of the original equation by solving the simple equation, so it is called the approximate analytic solution. This method can be used in mathematical physics of the original quantitative or qualitative analysis and discussion. At present, the theory for singular perturbation has been gradually established a number of effective methods, such as matched asymptotic expansion method, multi-variable expansion method, the boundary layer function method, V-L method, PLK method, WKB method, compensation compact method, multiple scales, KBM method, the average variational method, the constant field theory and the technique of diagonalization and so on. All sorts of methods for singular perturbation have been widely applied in many fields in natural science, which play a crucial role in solving practical problems. Most of dynamic mathematical models contain small parameters play a particularly important role in obtaining the uniformly valid asymptotic (approximate) solution for the complex nonlinear equations under the premise of being unable to get the accurate solution. In practical application, the numerical calculation and asymptotic methods are methods to find valid approximate solutions and complement each other.
The properties of the initial-boundary value problem for the 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 constructed. And then, using the method of multiple scales variables and the expanding theory of power series we seek solution of the problem. Finally, the existence, uniqueness and asymptotic behavior of the generalized solution for the problems are studied.
2. The nonlinear singularly perturbed Robin problems for reaction diffusion equations with boundary perturbation are considered in the paper. By the method of differential inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed under some suitable conditions.
3. Based on shishkin-type mesh, a singularly perturbed nonlinear neutral differential difference equation with negative shift initial boundary value problem is studied. B-spline collocation method and finite difference method is applied on the problem. Estimation of the error between exact solution and approximation solution is given.
引文
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