e
are considering the problem (
Parabolic
Dirichlet problem
).
Let
is the basis of
formed by the eigenfunctions of the operator
with the following
normalization:


(Definition of Galerkin basis 1)

The existence of such basis follows from the proposition
(
Eigenvalues of
symmetric elliptic operator
). To derive the orthogonality in
,
write
and integrate by parts on the LHS.
We form the
sum
and select the functions
to satisfy the
conditions
We integrate by parts the term
and write the equations for
in the
form


(Galerkin problem)

We substitute the definition of
and use the orthonormality of
.
We
obtain


(Galerkin coefficients)

This is a Cauchy system of ODEs and it has a form that insures that there is
always a solution
.
