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

本研究的对象是具有自由边界的偏微分方程（PDE）。 2001年，我们处理了R^2中单位正方形上定义的问题。由于自由边界是由通过某些特征值问题获得的潜在轮廓定义的，因此我们开发了偏微分算子上特征值问题的特征对的验证计算技术。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域，我们改进了现有的非凸多边形域上的偏微分方程的验证方法，得到了更简单和准确的方法。2002年，我们开发了一种圆环上定义的偏微分方程的验证计算方法。采用基于傅立叶-贝塞尔函数的谱法，需要保证精度的贝塞尔展开系数。通过涉及具有两点边界值的一维偏微分方程的特征值问题来定义系数。我们开发了验证特征值存在和不存在的方法，以获得系数的有效值。我们通过数值计算证明了这种不存在性的方法比现有方法更简单、更有效。此外，我们还开发了一个计算贝塞尔函数值的软件包，可以保证精度。它是在 INTLAB 上构建的，INTLAB 是一个在 MATLAB 上进行验证计算的区间计算库

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。