Mathematical Sciences: Uniform Asymptotic Approximations of Mathieu Functions

数学科学：Mathieu 函数的一致渐近逼近

• 批准号: 9104315
• 负责人: Timothy Dunster
• 金额: $ 3.7万
• 依托单位: San Diego State University Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-08-15 至 1994-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9104315&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Uniform; Asymptotic; Approximations

In this project the principal investigator will derive uniform asymptotic approximations for the solutions of second- order linear ordinary differential equations that contain a large parameter. The approximations would be uniformly valid in a domain of the complex plane that contains a coalescing turning point and a simple pole. In the beginning of the project the principal investigator will concentrate his efforts on approximating solutions of Mathieu's equation, including constructing realistic and explicit error bounds. He then hopes to extend his techniques and results to more general equations with turning points. Linear, second-order ordinary differential equations are in a real sense the building blocks for many complex theories in mathematical physics and asymptotic analysis. The solutions of such equations are useful in their own right as the solutions of interesting and important physical problems and equally as paradigms for more complicated phenomena. In this project the principal investigator will seek to find accurate approximate solutions of such an equation, known as Mathieu's equation, that arises in a number of applied contexts. Its solutions are known eponymously as Mathieu functions, and their properties are representative of asymptotic properties of solutions of more general equations involving turning-points.

在这个项目中，主要研究者将为包含大参数的二阶线性常微分方程的解推导一致渐近近似。 这些近似在包含合并转折点和简单极点的复平面域中一致有效。 在项目开始时，首席研究员将集中精力研究马蒂厄方程的近似解，包括构建现实且明确的误差范围。 然后，他希望将他的技术和结果扩展到更通用的带有转折点的方程。 线性二阶常微分方程实际上是数学物理和渐近分析中许多复杂理论的构建模块。 这些方程的解本身就可以作为有趣且重要的物理问题的解，并且同样可以作为更复杂现象的范例。 在这个项目中，首席研究员将寻求找到这种方程（称为马蒂厄方程）的精确近似解，该方程出现在许多应用环境中。 它的解与马蒂厄函数同名，它们的性质代表了涉及转折点的更一般方程的解的渐近性质。