Nonlinear Analysis by Numerical Verification Methods

数值验证方法的非线性分析

• 批准号: 13640105
• 负责人: YAMAMOTO Nobito
• 金额: $ 1.6万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640105/
• 关键词: verifind computation; spectrum method; nonlimear analysis; free boundary; 精密保証

The objects of this research are partial differential equations (PDEs) with free boundaries. In 2001, we treated problems defined on the unit square in R^2. As the free boundaries are defined by potential contour which is obtained through some eigenvalue problem, we developed techniques of verified computation for eigenpairs of eigenvalue problems on partial differential operators. In these techniques, we adopted spectrum methods for approximation and error estimationIn order to treat other shapes of domains than the unit square, we improved existing verification methods to PDEs on nonconvex polygonal domains and obtained a more simple and accurate methodIn 2002, we developed a methods of verified computation for PDEs defined on annuli. Using specrum method based on Fourier-Bessel functions, we needed coefficients of Bessel expansion with guaranteed accuracy. The coefficients are defined through an eigenvalue problem concerning a one-dimensional PDE with two points boundary valuesWe developed methods for verification of existence and nonexistence of eigenvalues in order to obtain the validated values of the coefficients. The method for nonexistence is simpler and more effective than existing methods, which we have shown by numerical calculationsMoreover, a software package to calculate the values of Bessel functions with guaranteed accuracy has been developed. It is constructed on INTLAB which is a library for interval calculation with verified computation on MATLAB

本文的研究对象是具有自由边界的偏微分方程。在2001年，我们处理了定义在R^2中单位平方上的问题。由于自由边界是由特征值问题得到的位势轮廓定义的，我们发展了偏微分算子特征值问题特征对的验证计算技术.为了处理单位正方形以外的其他形状的区域，我们改进了现有的非凸多边形区域上偏微分方程的验证方法，得到了一种更简单、更精确的方法。利用基于Fourier-Bessel函数的谱方法，需要保证精度的Bessel展开系数。11系数是通过一维两点边值偏微分方程的特征值问题来定义的，为了得到系数的有效值，我们提出了特征值存在性和不存在性的验证方法。13数值计算表明，这种方法比现有的方法更简单有效，而且还开发了一个计算Bessel函数值的软件包，保证了计算的精度。它是在INTLAB区间计算库上构建的，并在MATLAB上进行了验证计算

项目成果

Nakao, M.T., Watanabe, Y., Yamamoto, N.: "Verified numerical computations for an inverse elliptic eigenvalue problem with finite data"Japan Journal of Industrial and Applied Mathematics. Vo1.18. No.2. 163-173 (2001)

Nakao, M.T.、Watanabe, Y.、Yamamoto, N.：“用有限数据验证逆椭圆特征值问题的数值计算”日本工业与应用数学杂志。

Nakao, M.T., Watanabe, Y, Yamamoto, N: "Verified numerical computations for an inverse elliptic eigenvalue problem with finite data"Japan Journal of Industrial and Applied Mathematics. 18. 587-602 (2001)

Nakao, M.T.、Watanabe, Y、Yamamoto, N：“用有限数据验证逆椭圆特征值问题的数值计算”日本工业与应用数学杂志。

Nakao, M.T., Yamamoto, N.: "A guaranteed bound of the optimal constant in the error estimates for linear triangular element"(G.Alefelt, X.Chen (eds.)) Topics in Numerical Analysis With Special Emphasis on Nonlinear Problems, Computing Supplement 15, (Spri

Nakao, M.T.、Yamamoto, N.：“线性三角单元误差估计中最优常数的保证界限”（G.Alefelt、X.Chen（编辑））特别强调非线性问题的数值分析主题，

Yamamoto, N: "A simple method for error bounds of eigenvalues of symmetric matrices"Linear Algebra and its Applications. 324. 227-234 (2001)

Yamamoto, N：“对称矩阵特征值误差界的简单方法”线性代数及其应用。

Nakao, M.T., Yamamoto, N.: "A guaranteed bound of the optimal constant in the error estimates for linear triangular element Part II : Details"(U.Kulisch, R.Lohner, A.Facius (eds.)) Perspectives on Enclosure Methods, (Springer Wien) (New York). 265-276 (20

Nakao, M.T.，Yamamoto, N.：“线性三角形单元误差估计中最优常数的保证界限第二部分：详细信息”（U.Kulisch、R.Lohner、A.Facius（编辑））外壳透视