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组。2001年，我们处理了定义在R^2中单位平方上的问题。由于自由边界是由通过某些特征值问题得到的位势轮廓来定义的，我们发展了偏微分算子上特征值问题的特征对的验证计算方法。在这些技术中，我们采用谱方法进行逼近和误差估计。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域上偏微分方程组的验证方法，得到了一种更简单、更准确的方法。2002年，我们提出了一种定义在环上的偏微分方程组的验证计算方法。采用基于傅立叶-贝塞尔函数的谱方法，在保证精度的前提下，需要贝塞尔展开系数。系数的定义是通过一维两点边值问题的特征值问题来定义的。为了得到系数的确认值，我们发展了特征值的存在和不存在的验证方法。数值计算表明，这种不存在的方法比已有的方法更简单、更有效。此外，我们还开发了一个在保证精度的情况下计算贝塞尔函数值的软件包。它是建立在区间计算库INTLAB上的，并在MATLAB上进行了验证

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，

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

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。