Nonlinear Analysis by Numerical Verification Methods

本文的研究对象是具有自由边界的偏微分方程。在2001年，我们处理了在R^2的单位平方上定义的问题。由于自由边界是由某个特征值问题得到的势轮廓来定义的，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，为了处理除单位平方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。2002年，我们开发了一种环空上定义偏微分方程的验证计算方法。采用基于傅里叶-贝塞尔函数的谱方法，需要保证精度的贝塞尔展开系数。通过一个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，并提出了特征值存在性和不存在性的验证方法，以得到系数的验证值。通过数值计算表明，该方法比现有方法更简单、更有效，并开发了计算贝塞尔函数值的软件包。它是在INTLAB上构建的，INTLAB是一个区间计算库，并在MATLAB上进行了验证计算

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