Numerical Methods with Validation for Partial Differential Equations with Singularities

由于偏微分方程被用于各种自然现象的数学建模，因此发展偏微分方程的数值方法及其数值分析不仅是偏微分方程理论的一个重要课题，而且在物理、化学、生物、工程、金融营销等各个实践领域都是一个重要的课题。在该研究项目中，代表在一维两点边值问题和二维椭圆边值问题的研究方面取得了进展。提出了高阶数值方法，并对其进行了数值分析。

Since partial differential equations (PDE) are used to in mathematical modeling of various natural phenomena, the development of numerical methods and their numerical analysis for PDE is an important subject not only in the theory of PDE but also in the practical fields of physics, chemistry, biology, engineering, financial marketing, and so on. In this research project, the representative made progresses in the study of one-dimensional two-point boundary value problems and two-dimensional elliptic boundary value problems. Higher order numerical methods are proposed and numerical analysis of them is carried out.