Synthetical Researches on Applied Analysis and Computational Mathematics

应用分析与计算数学综合研究

基本信息

批准号：
03302009
负责人：
NISHIDA Takaaki
金额：
$ 8.58万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Co-operative Research (A)
财政年份：
1991
资助国家：
日本
起止时间：
1991 至 1992
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-03302009/
关键词：
Validated numerical analysis Free surface waves of fluid Finete element method Dimension reduction method Reaction Diffusion system Double exponential transformation Generating function of chaotic mapping Newton method 精度保証つき数値計算区間演算ソフト

项目摘要

New topics of applied analysis and computational mathematics in the wide sense are investigated with the collaboration of investigators. Research achievements are presented, discussed and evaluated in the two joint symposium on applied mathematics. Also several workshops and international conferences on selected topics are organized and partly supported by this project. These activated research communications and stimulated young mathematicians.1. The research group of head investigator obtained the following results:(1)Consolidation of the method of validated numerical computation. It gives a computer aided proof of the existence of periodic solution for the autonomous system of ordinary differential equations which depends delicately on the value of parameters. In the proof a software of interval analysis by the computer is necessary to evaluate the round-off errors at the each step of numerial computations. One software for the big computer Fujitsu M-1800 and one for Sun Workstations are completed and used in the researches.(2)Waves on the free surface of viscous fluid flowing sown on an inclined plane. A gloval in time existence theorem is obtained under appropriate conditions. Spectral analysis for the linearized system will be our next project which will require an improvement of the method of the above computer aided proof.2. Co-investigators obtained the following results:(1)Accurate error estimates for the solution of finite element method of evolution systems.(H.Fujita)(2)Analysis of structure change of the equilibrium solution over the wide range of parameter for reaction diffusion sytems by the reduction method of dimension.(M.Mimura)(3)Solution of boundary value problems by the double exponential transformation using Sine approximation.(M.Mori)(4)Approximation of a function by a generating function of a chaotic mapping.(M.Yamaguti)(5)Improvement of convergent radins of simplified Newton method.(T.Yamamoto)

在研究人员的合作下，研究了广义的应用分析和计算数学的新课题。两次应用数学联合研讨会对研究成果进行了展示、讨论和评价。该项目还组织了一些关于选定主题的研讨会和国际会议，并提供了部分支持。这些活动活跃了研究交流，激励了年轻数学家。 1．课题组组长取得了以下成果：（1）验证数值计算方法的巩固。它给出了计算机辅助证明常微分方程自治系统周期解的存在性，该系统微妙地依赖于参数值。在证明中需要使用计算机区间分析软件来评估数值计算每一步的舍入误差。完成了Fujitsu M-1800大型计算机的软件和Sun工作站的软件并用于研究。(2)倾斜平面上流动的粘性流体自由表面上的波。在适当的条件下得到了全局时间存在定理。线性化系统的频谱分析将是我们的下一个项目，这需要对上述计算机辅助证明的方法进行改进。 2．合作研究人员获得了以下成果：（1）对演化系统有限元法求解的误差进行了准确估计。（H.Fujita）（2）反应扩散参数范围内平衡解的结构变化分析(M.Mimura)(3)通过使用正弦近似的双指数变换解决边值问题。(M.Mori)(4)通过生成函数逼近函数混沌映射。(M.Yamaguti)(5)简化牛顿法收敛半径的改进。(T.Yamamoto)