Asymptotic Solution of a Model Non-linear Convective Diffusion Equation
R. E. Grundy · IMA Journal of Applied Mathematics · 1983
In this paper we consider the large-time solution of the equation for initial data with compact support. With m = 4 and n = 3 the equation models the flow of a thin viscous sheet on an inclined bed while for n ≯ m > 1 it has application in porous media flow under gravity. The equation can also be regarded as an analogue of Burgers equation in non-linear diffusion. It is known that two moving boundaries exist along which certain boundary conditions are required to hold. The paper extends earlier work and determines the analytic behaviour of the moving boundaries exist along which certain boundary conditions are required to hold. The paper extends earlier work and determines the analytic behaviour of the moving boundaries together with the structure of the solution which at large times is shown to depend crucially on the location in (n, m) parameter space.