text: The first computational model of dendritic solidification was published by Kobayashi, who used a phase-field model to solve two coupled partial differential equations describing the evolution of the phase-field,  (with  in the liquid phase and  in the solid phase), and the temperature field, , for a pure material in two dimensions:
  which is an Allen-Cahn equation with an anisotropic gradient energy coefficient:
  where  is an average value of ,  is the angle between the interface normal and the x-axis, and  and  are constants representing the strength and mode of anisotropy, respectively.
  The parameter  describes the thermodynamic driving force for solidification, which Kobayashi defines for a supercooled melt as:
  where  is a constant between 0 and 1,  is a positive constant, and  is the dimensionless equilibrium temperature. The temperature has been non-dimensionalized such that the equilibrium temperature is  and the initial temperature of the undercooled melt is . 
  The evolution equation for the temperature field is given by
  and is simply the heat equation with a source term due to the evolution of latent heat upon solidification, where  is a constant representing the latent heat normalized by the strength of the cooling.
  When this system is numerically evolved, random noise representing thermal fluctuations is introduced to the interface via the  term, where  is the magnitude of the noise and  is a random number distributed uniformly on .