Approximate solutions in capillary and chemical kinetics

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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.