The asymptotic solution of a family of boundary value problems involving exponentially small terms
The asymptotic solution of a family of boundary value problems involving exponentially small terms
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涉及指数小项的一族边值问题的渐近解
DOI：
10.1093/imamat/53.2.151
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发表时间：
1994
影响因子：
1.2
通讯作者：
H. R. Allen
中科院分区：
文献类型：
--
作者：
R. E. Grundy;H. R. Allen
本文考虑了边值问题族<$f（iv）-ff '“”+f' f“”= 0 f（0）= f '“（0）= 0 f（1）=1，fn（1）=-α}在极限|ε| ← 0.这个问题最近出现作为一个模型的磁场湮灭，但方程本身，各种不同的边界条件，有广泛的文献。采用渐近分析和数值分析相结合的方法，本文给出了一个全面的处理小|ε|问题，特别注意解决方案的对偶性问题。为|≥0时，这与渐近解中指数小项的出现密切相关。当α= 0（1）时，这些项z是由y = 1处的边界层所强迫的，处理这种情况所用的技巧在以前对方程的研究中是众所周知的。但是，对于小|α|的情况下，揭示了真正的性质的对偶性质的渐近解，这些众所周知的方法是不适用的，并通过初始值公式（*）的一种新的方法被使用。该方法是基于一个缩放方法，使问题减少到一个参数的家庭的初始值类型的问题。这大大简化了用于支持渐近分析的数值解的搜索和构造。当ε≤0时，收敛到ε=0解只发生在a的有限取值范围内，而当a足够小时，收敛到ε=0解的条件是：|ε|对于给定的边值问题只有一个解。
In this paper, the authors consider the family of boundary value problems ɛf(iv)-ff'''+f'f''=0f(0)=f''(0)=0f(1)=1, fn(1)=-α}in the limit |ε|← 0. This problem has recently appeared as a model for magnetic field annihilation but the equation itself, with variously different boundary conditions, has an extensive literature. Using a combination of asymptotic and numerical analyses, the paper gives a comprehensive treatment of the small |ε| problem, paying particular attention to the question of duality of solutions. For |≥0, this is intimately connected with the occurrence of exponentially small terms in the asymptotic solution. When α= 0(1) these termsz are forced by the boundary layer at y = 1, and the techniques used to deal with this case are well known from previous work on the equation. However, for small |α|, a case which reveals the true nature of the duality properties of the asymptotic solution, these well-known methods are not applicable, and a new approach via the initial value formulation of (*) is used. The approach is based on a scaling method which enables the problem to be reduced to a one-parameter family of problems of initial value type. This considerably simplifies the search for and construction of numerical solutions that are used to support the asymptotic analysis. For ε≤0, it is shown that convergence to the ε=0 solution only takes place for a restricted range of values of a and that, for sufficiently small |ε| there is only one solution to the given boundary value problem.