WIAS Preprint No. 779, (2002)

Second kind similarity solutions of the modified porous medium equation


  • Wagner, Barbara

2010 Mathematics Subject Classification

  • 76M60 76S05 76M45


  • Lie group methods, similarity solutions, asymptotics, porous medium flow


We consider the problem of a spreading ground water mound of liquid in a porous medium, situated on an impermeable horizontal solid layer. The mathematical formulation for this problem is given by the modified porous medium equation. We derive a global condition in form of an energy integral, describing the loss of liquid in the porous medium. This yields the necessary condition that determines the similarity exponents for the similarity solution of second kind, describing the long time behavior of the mound. We further apply our method to the problem when instead of an energy integral another conservation law, such as the first moment integral, obeyed by a family of antisymmetric solutions, is violated. Here, we consider as an application the problem of the impact of a flood infiltrating a porous medium. In all cases we will also solve and compare our analytical results with our numerical solution, and, if available, with examples existing in the literature.

Appeared in

  • J. Engrg. Math., 53 (2005) pp. 201-220.

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