e multiply the equation
with a smooth function
,
integrate over the domain
and apply the proposition (
Green formula
). We
arrive to the following weak formulation (see the section
(
Parabolic PDE section
) for review of
general theory):
Problem
(Heat equation weak formulation
2) Find the function
such
that
Note that we no longer require that the test functions
would satisfy the boundary condition. The boundary condition is now a part of
the weak formulation. Indeed, if we apply the proposition
(
Green formula
) to the problem
(
Heat equation weak formulation
2
) then we
get
The Neumann boundary condition is therefore called a "natural boundary
condition" and the Dirichlet condition is called an "essential boundary
condition".
