Approximate solutions in capillary and chemical kinetics

毛细管和化学动力学的近似解

• 批准号: 9345-2006
• 负责人: Siegel, David
• 金额: $ 0.66万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=364245
• 关键词: Approximate; solutions; capillary; chemical; kinetics

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.

由于大多数涉及微分方程的问题没有显式解，近似解可以提供许多信息和见解。这些都是我在毛细作用，化学动力学和Dirichlet问题研究中的特色。毛细表面是两种平衡流体之间的边界，例如液体在吸管。我已经开发了一个系统的方法产生近似解的径向对称问题，解除正确的体积。以前的结果本文用一种较简单的方法求出毛细管表面的近似解。这一点对于环形问题和外部问题是需要解决的。在1970年代，Paul Concus和Robert Finn发现了两个垂直平面形成的楔形体中液体的显著行为。当楔角足够小时，毛细管表面以一种精确的近似解所支配的方式变成无界的。我们用一种新的方法求出了毛细管表面的近似解。我们用一种新的方法求出了毛细管表面的近似解。我们用一种新的方法求出了毛细管表面的近似解。我们用一种新的方法求出了毛细管表面的近似解打算将这种分析扩展到尖点区域，例如，在存在接触垂直圆柱体的情况下的液体。这一领域有许多数学惊喜，数学导致了意想不到的物理见解。在化学动力学中，慢流形的概念已经由Simon Fraser和Marc Jersel引入并研究。这些提供了比准定常流形和非定常流形更上级的近似。状态或快速平衡近似。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实际有用的改进近似。Dirichlet问题是最古老和最重要的边值问题。多项式Dirichlet问题是在多项式方程给出的曲面上求出一个等于给定多项式的拉普拉斯方程的多项式解.这是一个基本问题.从更实际的观点来看，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.