Analysis and Computation of Nonlinear Structures

为了解决非线性现象的结构及其动力学问题，本文从流体动力学中的非线性结构、自由边界的动力学、断裂现象的科学计算、微分方程中出现的奇异点、分叉结构的跟踪等五个主要方面进行了研究。自动导引车周围的空气流动计算(Tabata)，轴对称流动问题的有限元分析(Tabata)，有限深度行波的分叉(Okamoto)，背景剪切流中的涡片卷曲(Okamoto)和具有单边边界条件的流动问题(Fujita)。两个呼吸子的模式选择(池田和西村)，行波脉冲的全局分叉图(池田)，高维空间中界面模式的存在性和稳定性(西村)，由反应扩散系统…导出的奇异极限系统Mimura)，广义曲线缩短方程的数值计算(TsutSumi)和多孔介质中的自由边界(Kawada)(2)。文(3)中弹塑性问题的任意线方法(Miyoshi)和改进的四节点四边形弯曲单元(Kikuchi)。(4)中退化抛物型方程(TsutSumi和Matano)的爆破问题。在(5)中，反应扩散系统和惯性流形(Morita)和Navier-Stokes沿着倾斜的平板(Nishida)流动。还研究了其他相关课题，例如脑动力学(Fujii)和混沌理论在社会科学中的应用(Yamaguti)。本项目的特点是数学分析与数值计算相结合。在数值方法的发展方面，取得了以下成果：改进的线性方程组共轭梯度法(Mitsui)，变量变换傅立叶积分的求积公式(Mori)，流体动力学方程分歧问题的计算机辅助证明(Nishida)和区域分解法收敛速度的区域依赖性(Fujita)。较少

For the purpose of solving the structure of nonlinear phenomena and their dynamics, the present research project has been dealt with the following five major fields : (1) Nonlinear structure in fluid dynamics, (2) Dynamics of free boundaries, (3) Scientific computing of fracture phenomena, (4) Singular points appearing in differential equations, (5) Tracking of bifurcation structure.Important results are as follows. Air flow computation around an automated guided vehicle (Tabata), Finite element analysis of axisymmetric flow problems (Tabata), Bifurcation of progressive waves of finite depth (Okamoto), Vortex sheet roll-up in the background shear flow (Okamoto) and Flow problems with unilateral boundary conditions (Fujita). Pattern selections for two breathers (Ikeda and Nishiura), Global bifurcation diagram of traveling pulses (Ikeda), Existence and stability of interfacial patterns in higher dimensional spaces (Nishiura), Singular limit systems derived from reaction-diffusion systems … More (Mimura), Numerical computation of generalized curve shortening equations (Tsutsumi) and Free boundaries in porous media (Kawarada) in (2). The arbitrary line method for elasto-plastic problems (Miyoshi) and Improved 4-node quadrilateral plate bending element (Kikuchi) in (3). Blow-up problems of degenerate parabolic equations (Tsutsumi and Matano) in (4). Reaction-diffusion systems and inertial manifolds (Morita) and Navier-Stokes flows down an inclined plate (Nishida) in (5). Other related topics have been also studied, Brain dynamics (Fujii) and Application of chaos theory to social sciences (Yamaguti) for instance.The present project is characterized by the mathematical analysis in cooperation with numerical computation. The following results have been obtained in the field of the development of numerical methods : An improved conjugate gradient method for large systems of linear equations (Mitsui), Quadrature formulas for Fourier type integrals obtained by variable transformation (Mori), Computer aided proofs for bifurcation problems in equations of fluid dynamics (Nishida) and The domain dependence of convergence rate of a domain decomposition method (Fujita) for instance. Less