This paper considers a family of one-dimensional nonlinear diffusion equations with absorption. In particular, the solutions that have interfaces that change their direction of propagation are examined. Although this phenomenon of reversing interfaces has been seen numerically, and some special exact solutions have been obtained, there was previously no analytical insight into how this occurs in the general case. The approach taken here is to seek self-similar solutions local to the interface and local to the reversing time. The analysis is split into two parts, one for the solution prior to the reversing time and the other for the solution after the reversing time. In each case the governing PDE is reduced to an ODE by introducing a self-similar coordinate system. These ODEs do not readily admit any nontrivial exact solutions and so the asymptotic behavior of solutions is studied. By doing this the adjustable parameters, or degrees of freedom, which may be used in a numerical shooting scheme are determined. A numerical algorithm is then proposed to furnish solutions to the ODEs and hence the PDE in the limit of interest. As examples of physical problems in which a PDE of this type may be used as a model the authors study the spreading of a viscous film under gravity and subject to evaporation, the dispersion of a population, and a nonlinear heat conduction problem. The numerical algorithm is demonstrated using these examples. Results are also given on the possible existence of self-similar solutions and types of reversing behavior that can be exhibited by PDEs in the family of interest.
|Journal||SIAM Journal on Applied Mathematics|
|Early online date||24 Jan 2012|
|Publication status||Published - 2012|
- nonlinear diffusion
- porous medium equation
- strong absorption
- slow diffusion