Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one
Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one
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具有一阶奇点的高阶线性微分方程的超渐近解
DOI：
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发表时间：
1998
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通讯作者：
A. Daalhuis
中科院分区：
文献类型：
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作者：
A. Daalhuis
对高阶齐次线性微分方程解的Poincare级数展开式中的余项，在秩为1的非正则奇点附近，给出了一系列的再展开式.这些重新展开式是一个级数，其项是斯托克斯乘数、原始庞加莱级数展开式的系数和某些多重积分（所谓的超端点）的乘积。该过程的每一步都将误差项的估计值减少一个指数小因子。本文的方法是基于Borel-Laplace变换，这使得它适用于其他问题。文末将该方法应用于鞍积分。此外，一个强大的新方法来计算斯托克斯乘子。包括一个数值例子。
A sequence of re–expansions is developed for the remainder terms in the well–known Poincare series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one. These re–expansions are a series whose terms are a product of Stokes multipliers, coefficients of the original Poincare series expansions, and certain multiple integrals, the so–called hyperterminants. Each step of the process reduces the estimate of the error term by an exponentially small factor. The method of this paper is based on the Borel–Laplace transform, which makes it applicable to other problems. At the end of the paper the method is applied to integrals with saddles. Also, a powerful new method is presented to compute the Stokes multipliers. A numerical example is included.