Study on asymptotic analysis of differential equations

A.复域微分方程的渐近分析内山分析了修正Bessel函数的无穷鞍积分表示，用超渐近分析、WKB方法和计算机辅助绘图，给出了Stokes几何新现象的全部例证。他还概述了斯托克斯现象的积分与鞍。大内研究线性偏微分方程在复杂的领域。假设非齐次项和一个解在超曲面K之外是全纯的，他证明了如果非齐次项在K附近有一定的渐近性质，则解也有相同类型的渐近性质。田原推广了经典的Maillet型定理，得到了如果存在形式解，则它属于某个Gevrey类。此外，他给唯一性定理，存在的全营养解决方案，存在奇点的解决方案。Yoshino具体计算了A. Voros使用Euler的Gamma函数的Binet公式。在真实的域中应用分析，平田利用光滑性质构造了小初值的时间全局解。斋藤构造了一个算法的绘制曲线定义为真实的零点的真实的代数多项式与真实的系数和实现它。后藤推广了一个新的建设的子因子Erlijman在算子代数。斋藤和小林有助于保持和加强计算机环境的研究。

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.