Let us consider non-linear heat conduction equation

_{} , (1)

where

_{}, (2)

with the initial condition _{} , _{} and boundary conditions

_{} , (3)

_{} At _{}. If _{} , condition (3) is the boundary condition of the second order, but if _{} , it is the boundary condition of the third order.

Let us introduce analytical grids in space and time:_{}, _{}, _{}, _{}.

Then, define grid functions _{} , _{} , _{} , _{} , _{} , _{} , _{} , _{}.