The analysis of initial-boundary-value problems using Hermite interpolation

In this paper we conduct an investigation into the feasibility of using Hermite interpolation as a practical tool for constructing polynomial approximations to initial-boundary-value problems for partial differential equations. The semi-analytic method uses Hermite interpolants to systematically estimate the time-dependent end point function values and/or derivatives which are not given by the boundary conditions or determined by the equations themselves. These estimates can then be used to analyse both the qualitative and quantitative structure of solutions. The idea is introduced via a series of examples intended to highlight various aspects of the method. These include a number of diffusion-convection-reaction models and examples involving unknown moving boundaries which illustrate how we can use the technique to identify such features as blow up regimes, steady states and similarity solutions together with their stability properties. (C) 2003 Elsevier Science B.V. All rights reserved.