【作者】 徐彪；

【导师】 陈松林；

【作者基本信息】 安徽工业大学 ， 应用数学， 2011， 硕士

【摘要】 奇异摄动理论及方法是一门应用非常广泛的学科,常用于求解非线性、高阶或变系数的数学物理方程的解析近似解,且对于摄动参数ε比较小的情况,经典的数值方法给不出令人满意的数值结果.目前关于奇异摄动方法的研究非常活跃且在不断拓展,许多奇摄动方法得到进一步发展,包括如匹配渐近展开法,多变量展开法,边界层函数法和多重尺度法.本文的内容如下:1、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.2、讨论了一类具有间断的拟线性奇异摄动边值问题,运用边界层函数法分段构造微分方程的形式渐近解,并用缝补法实现解的连续性,最后获得了解的存在性及渐近解的一致有效性.3、讨论了一类含时滞的奇异摄动边值问题,用边界层函数法和多重尺度法分段构造形式渐近解,并用缝补法把内部层连接起来,用微分不等式理论证明解的存在性,同时给出解的一致有效估计.更多还原

【Abstract】 The theory and method of application for singular perturbation problem is a very broad range of subject. The singularly perturbed method is used to find approximate analytical solutions of nonlinear, high order, or variable coefficients mathematical physical equations. Classical numerical methods usually give unsatisfactory numerical results when the singular perturbation parameterεis small. The current research is very active and constantly expanding .Recently, many approximate methods have been developed, including the method of matched asymptotic expansion, multi-variable expansion method, the boundary layer function method and the method of multiple scales.The main contents of this paper are outlined as follows:1. The nonlinear singularly perturbed Robin problems for reaction diffusion equations with boundary perturbation are considered in the paper. By using 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.2. A class of quasi-linear singularly perturbed boundary value problems with discontinuous of differential systems is considered. Using the method of boundary functions, the formal asymptotic expansions of the solution is constructed with step-wise method. At the same time, the solution is continuous under some suitable conditions. At last, the existence of the solution is proved and the uniformly efficiency of this asymptotic solution is obtained.3. A class of singularly perturbed boundary value problems with delay is considered. Using the method of boundary functions, the formal asymptotic expansions of the solution is constructed with step-wise method. Based on differential inequality techniques and the method of multiple scales variables, the existence of the solution is proved and the uniform validity of the asymptotic expansion is proved.更多还原