On the Asymptotic and Numerical Solution of Linear Ordinary Differential Equations
On the Asymptotic and Numerical Solution of Linear Ordinary Differential Equations
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线性常微分方程的渐近及数值解
作者：
A. Daalhuis;F. Olver
研究了任意阶齐次线性微分方程在单位秩奇点附近渐近解的性质。我们介绍了两种类型的解决方案的分类，显式和隐式。对于前者，存在一个扇区，当接近奇点时，其解被所有独立解所支配。没有这样的部门存在的隐式解决方案。因此，这两类解具有不同的唯一性。另一个区别是显式解的渐近展开式的误差界一般比隐式解的误差界强。
我们还研究了微分方程的数值积分的解决方案的计算。结果表明，它是例外，而不是规则的集成过程是稳定的，特别是隐式解决方案。为了克服这些不稳定性，我们开发的边界值方法，完成误差分析。
数值例子说明了显式和隐式解决方案的计算，以及相关的斯托克斯乘子。
An investigation is made of the nature of asymptotic solutions of homogeneous linear differential equations of arbitrary order in the neighborhood of a singularity of unit rank. We introduce a classification of the solutions into two types, explicit and implicit. For the former there exists a sector on which the solution is dominated by all independent solutions as the singularity is approached. No such sector exists for implicit solutions. In consequence, the two types of solution have different uniqueness properties. Another difference is that error bounds for the asymptotic expansions of explicit solutions are generally stronger than those for implicit solutions.
We also investigate the computation of the solutions by numerical integration of the differential equation. It is shown that it is the exception rather than the rule for the integration process to be stable, particularly so for implicit solutions. To overcome these instabilities we develop boundary-value methods, complete with error analysis.
Numerical examples illustrate the computation of both explicit and implicit solutions, and also the associated Stokes multipliers.