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We next use the variational statement derived previously to make a
discretization of the space dimension. Therefore we introduce a
partition
,
of the interval
into
subintervals of equal length
and let
be the corresponding space of continuous piecewise linear
functions vanishing at
. We now seek a solution approximation
such that for every fixed time
 |
(8) |
for all
.
Further, if
denotes a set of nodal basis functions of
we can expand
as
 |
(9) |
Notice that the coefficients
are time dependent but not
space dependent functions.
Substituting
into (7) and choosing test functions
,
we obtain
 |
(10) |
which is a
system of ordinary differential equations (ODE)
where the entries of the matrices
and
are given by
 |
(12) |
Here
denotes a vector holding the nodal values
of
.
Next: Time Discretization
Up: Time Dependent Problems
Previous: Variational Formulation
Mohammad Asadzadeh
2004-08-27