Remarks on application of different variables for the PKN model of   hydrofracturing. Various fluid-flow regimes
  The problem of hydraulic fracture for the PKN model is considered within the framework presented recently by Linkov (2011). The modified formulation is further enhanced by employing an improved regularized boundary condition near the crack tip. This increases solution accuracy especially for singular leak-off regimes. A new dependent variable having clear physical sense is introduced. A comprehensive analysis of numerical algorithms based on various dependent variables is provided. Comparison with know numerical results has been given. 
P. Kusmierczyk, G. Mishuris and M. Wrobel
physics.geo-ph math-ph math.MP math.NA