# Shooting procedure

The shooting method , even simple shooting method ( English (single) shooting method ), is a numerical method to boundary value problems of ordinary differential equations to be solved. The basic idea of ​​the method is to reduce the problem to the solution of an initial value problem .

The procedure is reminiscent of zeroing in artillery , a method of hitting a distant target with a projectile. The projectile is fired with a certain initial pitch. You vary this initial slope until you hit the target. Hence the name shooting procedure .

## Procedure

The boundary value problem of the second order with the function sought and on the right${\ displaystyle \, y (t)}$${\ displaystyle f}$

${\ displaystyle y '' (t) = f (t, y (t), y '(t)), \ quad y (t_ {1}) = a, \ quad y (t_ {2}) = b}$

is reformulated into an initial value problem

${\ displaystyle y '' (t) = f (t, y (t), y '(t)), \ quad y (t_ {1}) = a, \ quad y' (t_ {1}) = c }$

The second, unknown initial value can be freely selected. The initial value problem is integrated depending on the parameter c until the condition at the other edge is met. The solution of the initial value problem can be done with a numerical method, e.g. B. Runge-Kutta can be solved. depends on the initial value . Define a function F for this${\ displaystyle c}$${\ displaystyle \ quad y (t_ {2}) = b}$${\ displaystyle y (t \ ,; c)}$${\ displaystyle y (t \ ,; c)}$${\ displaystyle c}$

${\ displaystyle F (c): = y (t_ {2}, c) -b \ quad}$

This often non-linear system of equations can be solved numerically, for example with the Newton method or the bisection method . The solution of the initial value problem is a solution of the boundary value problem if and only if F has a zero in c :

${\ displaystyle 0 = y (t_ {2}, c) -b \ quad \ Leftrightarrow \ quad y (t_ {2}, c) = b}$

In practice, for reasons of stability, the so-called multi - target variant of the shooting method is used, in which solutions are calculated piece by piece in sub-intervals of a grid , from which the solution is then composed. ${\ displaystyle \ Delta = \ left \ {t_ {1} = x_ {0}, x_ {1}, \ dots, x_ {n-1}, x_ {n} = t_ {2} \ right \}}$${\ displaystyle [a, b]}$

## literature

• J. Stoer, R. Bulirsch : Introduction to Numerical Analysis. Springer, New York 1980
• A. Willers: Methods of practical analysis. 2nd Edition. De Gruyter, Berlin 1950
• L. Collatz: Numerical treatment of differential equations. Springer, Berlin 1951
• M. Hermann: Numerics of ordinary differential equations. Initial and marginal value problems , Oldenbourg Wissenschaftsverlag Munich and Vienna 2004, ISBN 3-486-27606-9