Mathematical Sciences: Harmonic Analysis Special Functions and Separation of Variables

数学科学：调和分析特殊函数和变量分离

• 批准号: 9400533
• 负责人: Willard Miller
• 金额: $ 6万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-04-01 至 1997-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400533&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Harmonic; Analysis; Special

Miller 9400533 This project continues a detailed mathematical study into the relationship between symmetries of the various fundamental partial differential and q-difference equations of mathematical physics; the possible coordinate systems in which these equations permit separation of variables and properties of the special function solutions which so arise. The work divides into several parts. The first concerns the problem of determining a practical definition of variable separation that applies to all (systems of) partial differential equations and the structure theory of possible separation types. Second is the problem of intrinsic characterization of separable coordinates, i.e., a coordinate- free characterization of coordinates. For Hamilton-Jacobi and Schrodinger equations this problem is essentially solved, but for systems of equations such as the Dirac and linear elasticity equations, it remains open, although progress has been made. Third, there is the problem of determining the special functions that arise when variables are separated in a particular equation and establishing the properties of these functions. Finally, there are general applications of these and related group theoretic methods to problems that arise in other science and engineering areas, such as quantum mechanics and radar/sonar. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. One particularly important form of approximation is through the use of special functions. The study of special functions related to solutions of partial differential equations lies at t he heart of this research project. ***

Miller 9400533这个项目继续对数学物理中各种基本偏微分方程式和Q-差分方程式的对称性之间的关系进行详细的数学研究；这些方程式允许分离变量的可能坐标系以及由此产生的特殊函数解的性质。这项工作分为几个部分。第一个问题是确定适用于所有偏微分方程组的变量分离的实用定义以及可能的分离类型的结构理论。第二个问题是可分坐标的内在刻画问题，即坐标的无坐标刻画。对于哈密顿-雅可比方程和薛定谔方程，这个问题基本上得到了解决，但对于狄拉克方程和线弹性方程等方程组，这个问题仍然是开放的，尽管已经取得了进展。第三，存在确定特定方程中的变量分离时出现的特殊函数以及确定这些函数的性质的问题。最后，这些群论方法和相关群论方法在其他科学和工程领域出现的问题上也有广泛的应用，如量子力学和雷达/声纳。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。一种特别重要的近似形式是通过使用特殊函数。与偏微分方程解相关的特殊函数的研究是本研究项目的核心。***