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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (6): 920-926.doi: 10.3969/j.issn.1005-3085.2015.06.013

• • 上一篇    下一篇

一类具有$n$阶转向点的大参数奇摄动方程的渐近解(英)

史娟荣1,2   

  1. 1- 安徽机电职业技术学院基础教学部，芜湖  241002
    2- 上海交通大学数学系，上海 200240
  • 收稿日期:2014-09-05 接受日期:2015-07-09 出版日期:2015-12-15 发布日期:2016-02-15
  • 基金资助:
    国家自然科学基金 (11202106)；安徽省教育厅自然科学基金 (KJ2015A418).

Asymptotic Solutions of Singularly Perturbed Equations for Large Parameter with Turning Point of $n$-th Order

SHI Juan-rong1,2   

  1. 1- Basic Teaching Department, Anhui Technical College of Mechanical and Electrical Engineering, Wuhu 241002
    2- Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240
  • Received:2014-09-05 Accepted:2015-07-09 Online:2015-12-15 Published:2016-02-15
  • Supported by:
    The National Natural Science Foundation of China (11202106); the Natural Science Foundation of the Education Deparment of Anhui Province (KJ2015A418).

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摘要： 本文研究了一类具有$n$阶转向点的大参数奇摄动方程解的渐近表达式．首先，利用Liouville-Green变换分别构造出当$n$为偶数和奇数情形下方程的外部解；随后，通过引入伸展变量，利用Bessel函数，我们分别构造出当$n$为偶数和奇数情形下方程在$n$阶转向点$x=0$附近的内层解；最后，我们利用匹配原理确定了外部解和内层解中的任意常数，从而得到方程的一致有效的一阶渐近表达式．

关键词: 转向点, Liouville-Green变换, Bessel函数, 奇异摄动问题

Abstract:

This paper considers the asymptotic solutions of a class of singularly perturbed equations for larger parameter with turning point of $n$-th order. Firstly, the outer solution when $n$ is odd or even, respectively, is obtained by using the Liouville-Green transformation. Then, the interior layer solution near the $x=0$ when $n$ is odd or even is constructed by introducing the stretching transformation and using the Bessel function. Finally, the arbitrary constants for the outer solution and interior layer solution are determined by using the matching principle. Thus, we obtain the uniformly valid asymptotic expression of the equation.

Key words: turning point, Liouville-Green transform, Bessel function, singular perturbation problem

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