The application of Hermite interpolation to the analysis of non-linear diffusive initial-boundary value problems

In this paper we examine the feasibility of using two-point Hermite interpolation as a systematic tool in the analysis of initial-boundary value problems for non-linear diffusion equations. We do this by considering a series of examples for the porous medium equation involving both fixed and moving boundaries. Essentially, the idea is to construct polynomials which fit the known and unknown function values and their derivatives at the two end points of a given interval. Systems of ordinary differential equations are then obtained for the unknown end point functions of time which need to be determined in order to specify the polynomial representation-the initial conditions for such systems are related to the initial data for the original problem. As well as constructing approximate solutions, it emerges that the method is particularly useful in identifying steady states and similarity solutions together with their stability and other asymptotic properties. We believe that the technique provides scientists and applied mathematicians with a valuable strategy in the analysis of the qualitative and quantitative features of solutions to initial-boundary value problems involving non-linear diffusion and related equations.