3.2.3
When the data d is available at discrete points: 
, 
, the Euler equation for the adaptive string is
where 
is the Dirac delta function. Integrating the
above equation once yields
where c is a constant. From the above, the solution f is determined by the following
with appropriate boundary conditions at 
and 
.
Obviously, f in this case is piecewise 
continuous; more exactly,
it is composed of consecutively joined line segments.
The adaptive string model can be extended to 2D and higher order equivalents. The 2D adaptive membrane has the following
corresponding to the Euler equation (3.25), where data
and solution 
represent images. Extending the DA to
second order, one obtains the following for the adaptive rod on 1D
and for the adaptive plate on 2D
