Harmonic Analysis and Numerical Analysis on Functional Defferential Equations and Partial Differential Equations

泛函微分方程和偏微分方程的调和分析和数值分析

• 批准号: 11640155
• 负责人: NAITO Toshiki
• 金额: $ 2.3万
• 依托单位: The University of Electro-Communications
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640155/
• 关键词: Functional Differential Equations; Variation-of-constants Formula; Periodic Solutions; Almost Periodi Solutios; Spectrum; Finite Element Method; Perfect Fluid; Domain Decomposition; 差分方程式; 不動点定理; ヴォルテラ型方程式; 半群; 全安定性; 極限方程式; フレッドホルム作用素; 安定性

1. The results on the analytic investigation of solution spaces, stability, existence of peridic(P-) or almost periodic(AP-) solutions in functional, partial or stochastic differential equations(DE's) : We studied the Fourier-Carleman spectrum of bounded solutions of linear DE's. Main result are decompositions of bounded solutions corresponding to the separation of spectrum, its application to the existence of P- or AP-solutions and admissiblility of function spaces. Difference equations are treated similary. Fundamental results are given on the existence and uniqueness of solutions to functional differential equations (FDE's) in Banach spaces. The spectral theory of solution semigroups of linear FDE's are applied to stability and existence of P-solutions. A new variation-of-constants formula for FDE's are established on the abstract phase space; the formula are shown to be effective on the decomposition of the phase space and the existence of P- or AP solutions.2. The results on the numerical analysis on the example of concrete applications and the development of the technique of numerical computation: About the finite element method(FEM) on 2 dimensional perfect fluid around a wing, we studied the numerical construction of conformal function of a wing by using the results on FEM to the discrite version of Laplace problem in the interior of a disk. About the scattering problem in unbounded region, we studied the numerical solving manner by using 'the domain decompositon method as well as the fictitious domain method. Among them we obtained a new view of numerical treatments of Dirichlet-Neumann problem. Including the FEM approximation of mixed type to Poisson equation, we observed the importance of the essential spectrum in the study of FEM.

1.本文研究了泛函微分方程、偏微分方程和随机微分方程的解空间、稳定性、周期解（P-）和概周期解（AP-）的存在性。主要结果是对应于谱分离的有界解的分解，它在P-或AP-解的存在性和函数空间的可容许性方面的应用。差分方程的处理类似。给出了Banach空间中泛函微分方程解的存在唯一性的基本结果。利用线性FDE解半群的谱理论研究了P-解的稳定性和存在性。在抽象相空间上建立了FDE的一个新的常数变分公式，并证明了该公式对相空间的分解和P-或AP解的存在性是有效的.具体应用实例的数值分析结果及数值计算技术的发展：关于二维理想流体绕翼问题的有限元方法，我们将有限元方法的结果应用于圆盘内部离散形式的拉普拉斯问题，研究了机翼保角函数的数值构造。对于无界区域内的散射问题，研究了区域分解法和虚拟区域法的数值求解方法。其中我们得到了Dirichlet-Neumann问题数值处理的一个新观点。通过对Poisson方程的混合型有限元逼近，说明了本质谱在有限元研究中的重要性。

项目成果

Y.Hino, T.Naito, N.V.Minh, J.S.Shin: "Almost Periodic Solutions of Differential Equations in Banach Spaces"Taylor & Francis. 250 (2002)

Y.Hino、T.Naito、N.V.Minh、J.S.Shin：“Banach 空间中微分方程的几乎周期解”Taylor

KAKO, Takashi and NASIR, H. Mohamed: "Essential spectrum and mixed type finite element method"Lecture Notes in Computational Scikence and Engineering, Mathematical Modeling and Numerical Simulation in Continuum Mechanics, Proceedings of International Symp

KAKO, Takashi 和 NASIR, H. Mohamed：“本质谱和混合型有限元法”计算科学与工程、连续介质力学中的数学建模和数值模拟讲义，国际 Symp 会议录

Satoru Murakami, Toshiki Naito and Nguyen Van Minh: "Evolution semigroups and sums of commuting operators : A new approach to the addmissibility theory of function spaces"J. Differential Equations. 164. 240-285 (2000)

Satoru Murakami、Toshiki Naito 和 Nguyen Van Minh：“演化半群与交换算子之和：函数空间可容性理论的新方法”J.

K.Yamaguchi: "Spaces of polynomials with real roots of bounded multiplicity"to appear J. Math. Kyoto Univ.. 42. (2002)

K.Yamaguchi：“具有有界重数实根的多项式空间”出现在 J. Math 中。

Toshiki Naito, Jong Song Shin and Nguyen Van Minh: "Periodic solutions of linear differential equations"RIMS Kokyuroku. 1216. 78-89 (2001)

Toshiki Naito、Jong Song Shin 和 Nguyen Van Minh：“线性微分方程的周期解”RIMS Kokyuroku。