n context
of the condition (
Generic parabolic
PDE setup
) we introduce partition of the interval
:
We define
to be the
class
We introduce the
notations
for
.
We aim to approximate the solution
of the problem
(
Weak formulation
with respect to time parameter
) with a function from
for fixed
and vanishing
.
We calculate for
and
:
Note that
We
continue
The equation (
Timeweak problem
) transforms
into
for any
.
Thus,
.
We replace
with
:
We can switch former problem for the latter if we can prove later that both
problems have a unique solution. We merge conditions and arrive
to
At this point we fix
and take a
with support within
.
We arrive to the following problem.
Note that the solution of the equation
(
Generic parabolic PDE problem
)
satisfies the above problem.
