except for the first and last subdomains, where in the Neumann step the Neumann conditions are replaced by homogeneous Dirichlet conditions
at the physical boundaries.
Schaefer, "Lower bounds for blow-up time in parabolic problems under Dirichlet conditions
," Journal of Mathematical Analysis and Applications, vol.
He proposed that the homogeneous Dirichlet conditions
may be satisfied exactly by representing the solution as the product of two functions: (1) an real-valued function that takes on zero values on the boundary points; and (2) an unknown function that allows to satisfy (exactly or approximately) the differential equation of the problem.
Considering the Dirichlet conditions
at x = 0 and x = [x.
This field satisfies the Dirichlet conditions
at all four sides.
where [GAMMA] symbolizes the boundary, and the Dirichlet conditions
are considered at 20[degrees]C for natural convective air-cooling BEM formulation, Green's function as a solution of the Equation (2) through (7) as follows.
The inhomogeneous polyanalytic equation is studied by Begehr and Kumar  in D with Dirichlet conditions
and the following result is obtained.
The system (11) is accompanied with the following time-independent Dirichlet conditions
1], these functional conditions include as a particular case, the Dirichlet conditions
In this case, the profile record builds sets of (x, y) points for each numerical method, where the x values are grid points, and the y values are relative elapsed time changes for mixed boundary conditions with respect to Dirichlet conditions
, changes in elapsed time for Neumann conditions with respect to Dirichlet conditions
, and relative changes in error for derivative conditions with respect to Dirichlet conditions
are assumed on a closed subset [[GAMMA].
The Robin condition is consistent with the analysis in  where it is shown that Dirichlet conditions
on inflow are preferred as v approaches 0.