Study on asymptotic analysis of differential equations

A.复域微分方程的渐近分析。Uchiyama分析了修正贝塞尔函数的无限鞍的积分表示，用超渐近分析、WKB方法和计算机辅助图形学给出了Stokes几何的所有新现象。他还概述了带鞍积分的Stokes现象。大内研究复域的线性偏微分方程。假设非齐次项和解在超曲面K外是全纯的，证明了如果非齐次项在K附近具有一定的渐近性质，则解具有同类型的渐近性质。Tahara推广了经典的Maillet类型定理，得到如果存在一个形式解，它就属于某一类Gevrey。进一步给出了唯一性定理，全胚解的存在性，解的奇异点的存在性。吉野用欧拉函数b的Binet公式具体计算了a . Voros提出的谐振子的Jost函数。实域应用分析。Hirata利用平滑特性构造了小初始数据的时间全局解。Saito构造了一种绘制实数系数的实数代数多项式的实数零点曲线的算法，并实现了该算法。Goto用Erlijman在算子代数中推广了一种新的子因子构造。齐藤和小林帮助保持和加强了研究的计算机环境。

A. Asymptotic analysis of differential equations in complex domain.Uchiyama analyzed an integral representation with infinite saddles of the modified Bessel functions to give all illustration of new phenomena of Stokes geometry by hyperasymptotic analysis, WKB method and computer aided graphics. He also gave an overview of Stokes phenomenon of integrals with saddles. Ouchi studied linear partial differential equations in the complex domain. Supposing that the inhomogenious term and a solution are holomorphic outside of a hypersurface K, he proved that if the inhomogenious term has a certain asymptotic property near K, the solution has the asymptotic property of the same type. Tahara extended the classical Maillet type theorem to obtain that if a formal solution exists, it belongs to a certain Gevrey class. Furthermore, he gave uniqueness theorems, existence of a holomrophic solution, existence of singular points of the solution. Yoshino computed concretely the Jost function of a harmonic oscillator proposed by A. Voros using the Binet formula for Euler's Gamma functionB. Applied analysis in real domain.Hirata constructed a time global solution for small initial data using smoothing property. Saito constructed an algorithm of drawing of curves defined as real zeros of real algebraic polynomials with real coefficients and implemented it .Goto generalized a new construction of subfactors by Erlijman in operator algebra.Saito and Kobayashi helped to keep and enhance the computer environments for research.