Approximate solutions in capillary and chemical kinetics

由于大多数涉及微分方程的问题都没有显式解，所以近似解可以提供很多信息和洞察力。这些都是我在毛细作用、化学动力学和狄里克莱特问题研究中的特点。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，这些问题提升了正确的体积。以前的结果可以以更简单的方式获得，并且可以获得更好的近似。这需要针对环状和外部问题来解决。例如，在20世纪70年代，S、保罗·孔库斯和罗伯特·芬恩发现了由两个垂直平面形成的楔形液体的非凡行为。当楔形角度足够小时，毛细管表面以一种精确的近似解的方式变得无界。我们打算将这种分析扩展到尖端区域，例如，在接触垂直圆柱体的情况下，液体。在这一领域，有许多数学上的惊喜，数学导致了意想不到的物理洞察。在化学动力学方面，慢流形的概念是由Simon Fraser和Marc Roussel引入并研究的。它们提供了比通常使用的准稳态或快速平衡近似更好的近似。我们一直在澄清围绕慢流形的数学问题和计算它们的迭代方法。这项工作有可能导致改进的实用逼近。Dirichlet问题是最古老也是最重要的边值问题。而多项式Dirichlet问题是找到拉普拉斯方程的多项式解，它等于在给定的曲面上的给定多项式一个多项式方程。这是一个基本感兴趣的问题。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Dirichlet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.