Mathematical Sciences: Uniform Asymptotic Approximations of Mathieu Functions

在这个项目中，首席研究员将获得 第二类问题解的一致渐近近似 阶线性常微分方程，包含一个大的 参数. 这些近似在一个 域的复平面，其中包含一个合并转向 点和一个简单的杆。 在项目开始时， 首席调查员将集中精力 Mathieu方程的近似解，包括 建立现实的和明确的误差界限。 他希望， 将他的技术和结果推广到更一般的方程 有转折点。 线性二阶常微分方程在 在真实的意义上， 数学物理和渐近分析。 的解 这样的方程本身是有用的，因为它们是 有趣而重要的物理问题， 更复杂现象的范例。 本工程 首席研究员将寻求找到准确的近似值 这种方程的解，被称为马蒂厄方程， 出现在许多应用环境中。 其解决方案是已知的 Mathieu函数，它们的性质是 解的渐近性质的代表 包含转折点的一般方程。

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.