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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 : .
Quasi-linearization:
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. .

* Quasi-linearization: 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