e assume that the condition
(
Generic parabolic PDE setup
)
takes place. We further assume that the operator
is
independent.
We extend the function
to the entire
line
and apply the Laplace transform (see the section
(
Laplace transform
)) to the
equation
and
obtain
Thus
where, according to the properties of
listed in the condition (
Generic
parabolic PDE setup
), the resolvent
has poles on the positive side of the real axis. According to the section
(
Laplace transform
) and for
,
Then we want to transform the contour of integration
to give the term
exponential decay of order
to allow for GaussHermit quadrature, see the formula
(
GaussHermite Integration
). Such
operation would impose analyticity requirements on
and thus, decay requirements on
and would depend on positioning of poles of
.
Let us assume that we can transform the contour into
without crossing any singularities of
.
Then we arrive
to
and, after application of the formula
(
GaussHermite
Integration
),
for some complex numbers
and elements
.
Thus we reduced the original problem to solving several spacial elliptic
problems
that may be executed in parallel.
Transformation of the contour may be unnecessary if the function
already have the exponential decay.
