一个具有无限长区域的奇摄动边值问题

A Singular Perturbation Boundary Value Problem with Unbounded Interval

利用匹配渐近展开法,讨论了一个具有无限长区域的奇摄动边值问题.首先直接求得了问题外部解的渐近表达式,其次由收缩变换求出了问题的内层解,再用匹配原理对外部解和内层解进行匹配,得到了问题的一致有效的渐近解,即一阶渐近展开式.最后揭示了用不同于常见的渐近序列进行求解,可以得到一致的结果.此问题的可解性进一步说明了用这种方法解决此类问题的可行性.

Using the method of matched asymptotic expansions, a singular perturbation boundary value problem with unbounded interval is discussed. Firstly, the asymptotic expression of the outer solution for the problem is solved. Secondly, the inner solution is given by the contraction transformation. And by the matched principle, matching the outer solution and the inner solution, the uniformly valid asymp- totic solution, i.e. first - order asymptotic expansion of the solution for the problem is obtained. Final- ly, it is revealed that if the problem is resolved using the asymptotic sequence different from the common one, the consistent results can be obtained. Solvability of this problem further illustrates the feasibility of solving this class of problems using this method.