Approximate solutions in capillary and chemical kinetics

由于大多数涉及微分方程的问题没有显式解，近似解可以提供很多信息和洞察力。这些都是我研究毛细现象、化学动力学和狄利克雷问题的特点。毛细管表面是处于平衡状态的两种流体之间的边界，例如吸管中的液体。我已经开发了一种系统的方法来生成径向对称问题的近似解，从而提升正确的体积。以前的结果可以用更简单的方法得到，并且可以得到更好的近似。这需要解决环形和外部问题。在20世纪70年代，保罗·孔库斯和罗伯特·芬恩发现了液体在由两个垂直平面组成的楔子中的非凡行为。当楔角足够小时，毛细管表面以一种精确的近似解控制的方式变得无界。我们打算将这种分析扩展到尖端区域，例如，液体在接触垂直圆柱体的情况下。这个领域有很多数学上的惊喜，数学也带来了意想不到的物理见解。在化学动力学中，慢流形的概念是由西蒙·弗雷泽和马克·鲁塞尔提出并研究的。这些提供了比准稳态或通常使用的快速平衡近似更好的近似。我们一直在阐明围绕慢流形的数学问题以及计算慢流形的迭代方法。这项工作有可能导致实际用途的改进近似。Dirichlet问题是最古老也是最重要的边值问题。多项式狄利克雷问题是求拉普拉斯方程的多项式解它等于一个给定的多项式在一个多项式方程给出的曲面上。这是一个基本利益问题。从更实际的角度来看，通过用多项式逼近边界和边界数据，这种方法可以得到更一般的狄利克雷问题的近似解。

Since most problems involving differential equations do not have explicit solutions, approximate solutions can provide much information and insight.  These are featured in my research on capillarity, chemical kinetics and the Dirichlet problem.  A capillary surface is the boundary between two fluids in equilibrium, e.g. liquid in a straw.  I have developed a systematic way of generating approximate solutions to radially symmetric problems which lift the correct volume.  Previous results can be obtained in a simpler way and better approximations can be obtained.  This needs to be worked out for the annular and exterior problems.  In the 1970's Paul Concus and Robert Finn discovered the remarkable behaviour of liquid in a wedge formed by two vertical planes.  When the wedge angle is small enough the capillary surface becomes unbounded in a way governed by a precise approximate solution.  We intend to extend this analysis to  cusp regions, e.g. liquid in the presence of touching vertical cylinders.  This area has had many mathematical surprises and the mathematics has led to unexpected physical insights.  In chemical kinetics, the concept of slow manifold has been introduced and studied by Simon Fraser and Marc Roussel.  These provide a superior approximation than either the quasi-steady-state or the rapid equilibrium approximations commonly used.  We have been clarifying the mathematical issues surrounding slow manifolds and the iterative methods for computing them.  This work has the potential to lead to improved approximations of practical usefulness.  The Dirichlet problem is the oldest and most important boundary value problem.  The polynomial Dirichlet problem is to find a polynomial solution to Laplace's equation which is equal to a given polynomial on a surface given by a polynomial equation.  This is a problem of basic interest.  From a more practical point of view, by approximating a boundary and the boundary data by polynomials, this approach can lead to approximate solutions to a more general Dirichlet problems.