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 
, 
, 
, 
, 
, 
, 
, 
.