Co-Operative Research of Numerical Analysis of Nonlinear Problems

非线性问题数值分析的合作研究

• 批准号: 02302012
• 负责人: SHINOHARA Yoshitane
• 金额: $ 3.46万
• 依托单位: Tokushima 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: 1990
• 资助国家: 日本
• 起止时间: 1990 至 1991
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-02302012/
• 关键词: quasiperiodic phenomena; arc-length method; bifurcation; optimum shape design; finite element analysis; electromagnetism; parallel computation; dynamics of phase and surface; 非線形現象の数理; 常微分方程式の数値計算; 偏微分方程式の数値計算; 分岐現象; 流体力学の数値解析; 構造物の数値解析; 化学反

1. Head Investigator's results : (1) Existence and uniqueness theorem with an error estimation of Galerkin approximation has been established. This theorem is very useful for numerical analysis of quasiperiodic phenomena which appear in nonlinear oscillation, modulation and detection of communication. The theorem is applied to compute the quasiperiodic solutions to Van der Pol type and Duffing type equations. (2) The concept of numerically ill conditioning of solutions of the initial value problem of ordinary differential equations is given. The concept is novel and important in numerical analysis of ordinary differential equations. (3) The arc-length method (or geometric method) becomes very powerful for the bifurcation analysis in nonlinear problems.2. Investigators' main results : (1) T. Ushijima investigates the eigenvalue problems of water waves and gives some important paterns of global behaviour of relative erros in eigenvalues. (2) H. Okamoto examines the stabilities of steady … More axisymmetric flows and determines how these stabilities change upon varying the geometric parameters in 2-dimensional Navier-Stokes equations. (3) , H. Kawarada studies on Stefan problems with free boundary and computes the solidification problems with change of volume. The short time existence of the unsteady free boundary appearing in the porous media is studied by use of Nash-Moser's implicit function theorem. Nash-Moser's implicit theorem has been modified in an applicable form. (4) F. Kikuchi studies the mixed formulations for finite element analysis in electromagnetism and solid-state mechanics. He developed some weak formulations for finite element analysis of magnetostatic and electrostatic problems by means of the Hilbert space method. He received the ISHIKAWA award. (5) T. Mitsui made a new A-stable 3-stage fourth order Runge-Kutta formula which can be calculated in parallel. Specifically, forcus is given to general solution for formulae parameters of IRK under the symplectic and the order conditions. Examples of such formulae are constructed and linear orders are given for up to three stages. (6) M. Mimura studies the dynamics of phase and surface of chemical waves and solidification by means of the moving pictures. A size-space distribution model of biological individuals including two effects of density-dependent growth rates for size and chemotactic aggregation for space have been proposed. Assuming that the spatial movement is rapid in comparison with the growth process, he uses time-scaling arguments to reduce the model to an approximating system of only size distribution. He attended and presented a lecture on ICIAM (1991, USA) about his work on pattern formation arising from systems of reaction diffusion equations in space two and higher. Less

1. 首席研究员的成果： (1) 建立了伽辽金近似误差估计的存在唯一性定理。该定理对于通信中非线性振荡、调制和检测中出现的准周期现象的数值分析非常有用。 Van der Pol 型和 Duffing 型方程的准周期解 (2) 初值解的数值病态概念。给出了常微分方程的问题。这个概念在常微分方程的数值分析中是新颖且重要的。 (3) 弧长方法（或几何方法）对于非线性问题的分岔分析变得非常强大。2．主要结果：（1）T. Ushijima 研究了水波的特征值问题，并给出了特征值中相对误差的全局行为的一些重要模式（2）H. Okamoto 检查了稳定的稳定性……更多轴对称流动并确定这些稳定性如何随二维纳维-斯托克斯方程 (3) 中的几何参数变化而变化，H. Kawarada 研究自由边界的 Stefan 问题并计算随体积变化的凝固问题。利用 Nash-Moser 隐函数定理对多孔介质中出现的不稳定自由边界进行了研究，并以适用的形式对 Nash-Moser 隐函数定理进行了修改。 F. Kikuchi 研究了电磁学和固态力学中有限元分析的混合公式，他通过希尔伯特空间方法开发了一些用于静磁和静电问题的有限元分析的弱公式。他获得了石川奖。 T. Mitsui提出了一种新的可并行计算的A稳定三级四阶龙格-库塔公式，重点研究了IRK下公式参数的通解。构造了此类公式的示例，并给出了最多三个阶段的线性顺序。 (6) M. Mimura 通过 A 尺寸的图像研究了化学波和凝固的相和表面动力学。 -提出了生物个体的空间分布模型，包括尺寸的密度依赖性生长速率和空间的趋化聚集两种效应，假设空间运动与生长过程相比是快速的，他使用时间尺度参数来减少。模型到他参加了 ICIAM（1991 年，美国）的讲座，介绍了他在二维及更高空间中的反应扩散方程组中的模式形成的工作。

项目成果

篠原 能材: "数理解析研究所講究録 連立非線形方程式の大域における数値解法とその応用" 京都大学数理解析研究所, (1991)

野崎筱原：“数学科学研究所的Kokyuroku：联立非线性方程的全局数值解及其应用”京都大学数学科学研究所，（1991）

H. OKAMOTO and S. J. TAVENER: "Degenerate O (2)-equivalent bifurcation equations and their application to the Taylor problem" Japan J. Indust. Appl. Math.8. 245-273 (1991)

H. OKAMOTO 和 S. J. TAVENER：“简并 O (2) 等价分岔方程及其在泰勒问题中的应用”Japan J. Indust。

Y. SHINOHARA and A. KOHDA: "Numerical analysis of the quasiperiodic solutions to Duffing type equations" J. Math. Tokushima Univ.26. (1992)

Y. SHINOHARA 和 A. KOHDA：“Duffing 型方程准周期解的数值分析”J. Math。

篠原 能材: "数理解析研究所講究録748「連立非線形方程式の大域における数値解法とその応用」" 京都大学数理解析研究所, 117 (1991)

野崎筱原：“数学科学研究所Kokyuroku 748“联立非线性方程的全局数值解及其应用””京都大学数学科学研究所，117（1991）

H.OKAMOTO and S.J.TAVENER: "Degenerate O(2)-equivalent bifurcation equations and their application to the Taylor problem" Japan J.Indust.Appl.Math.8. 245-273 (1991)

H.OKAMOTO 和 S.J.TAVENER：“简并 O(2) 等价分岔方程及其在泰勒问题中的应用”Japan J.Indust.Appl.Math.8。