Synthetical Researches on Applied Analysis and Computational Mathematics

应用分析和计算数学在广义上的新课题进行调查的合作调查。本文介绍了两次应用数学联合学术讨论会的研究成果，并对其进行了讨论和评价.此外，该项目还组织并部分支助了若干关于选定专题的讲习班和国际会议。这些激活了研究交流并激励了年轻的数学家。1.本课题组取得了以下成果：（1）巩固了已验证的数值计算方法。本文给出了一个精确依赖于参数值的自治常微分方程组周期解存在性的计算机辅助证明。在证明过程中，需要用计算机编制区间分析软件，对每一步数值计算的舍入误差进行估计。在此基础上，开发了一套适用于大型计算机Fujitsu M-1800的软件和一套适用于Sun工作站的软件。（2）粘性流体在斜面上自由表面上的波。在适当的条件下，得到了一个全局时存在定理。线性化系统的频谱分析将是我们的下一个项目，这将需要改进上述计算机辅助证明的方法。得到了以下结果：（1）发展系统有限元方法解的精确误差估计。（2）用降维法分析反应扩散系统平衡解在宽参数范围内的结构变化。（3）用双指数变换和正弦逼近解边值问题。（M.Mori）（4）混沌映射的生成函数对函数的逼近。（5）简化牛顿法收敛半径的改进。（山本太郎）

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)