The PDE
is converted to first order by introducing a new variable which is defined by
Use of this new variable in the input system results in the first order system:
As a further example, for a PDE containing a fourth time derivative of a dependent variable, three new variables need to be introduced to bring the system to first order.
Consider the following PDE, initial conditions, and boundary conditions (the linear biharmonic equation for a bar with rigidly fixed ends):
The automatic approach reformulates this input as follows.
Alternatively, you can reformulate this system as follows:
which may be a good choice as all boundary conditions are of the 'point'='value' form (so there will be no error introduced by discretization).
As yet another alternative:
which may be a good choice because the system has the lowest possible maximal differential order in .
Compare the error of the solutions obtained from these equivalent systems over the range for fixed step sizes.
Note: Use a for which you have an approximate solution.
Now using a different approximate solution: