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4.3.1 Shooting Method
 Transform the given boundary value problem
into an initial value problem with estimated parameters
 Adjust the parameters iteratively to reproduce the given boundary values
 First trial shot:
 Augment the boundary values given at
by
estimated parameters
to obtain an IVP. Integrate numerically up .
The newly calculated values of at ,
will in general deviate from the given boundary values
. The difference vector
is stored for further use.
 Second trial shot:
 Change the estimated initial values
by some small amount,
,
and once more integrate up to . The values
thus obtained are again different from the required
values :
.
 Quasilinearization:
 Assuming that the deviations
and
depend linearly on the estimated
initial values
and
, compute that vector
which would make the deviations disappear:
Iterate the procedure up to some desired accuracy.
EXAMPLE:
* First trial shot: Choose
.
Applying 4th order RK with
we find
. Thus
.
* Second trial shot: With we find
, i.e. .
* Quasilinearization: From
we find
.
Iteration: The next few iterations yield the following values for
and
:



3 
0.405 
 0.041 
4 
0.440 
0.003 
5 
0.437 
0.000 
Next: 4.3.2 Relaxation Method
Up: 4.3 Boundary Value Problems
Previous: 4.3 Boundary Value Problems
Franz J. Vesely Oct 2005
See also: "Computational Physics  An Introduction," KluwerPlenum 2001