The Shooting Method, Nonlinear Case

Similar to linear case, but cannot piece solution as linear combination of 2 IVP. Instead, we create a sequence of IVP's of the form

$$\left\{\begin{array}{l} y''=f(x,y,y') x_0\le x\le b y(x_0)=\alpha y'(x_0)=t \end{array}\right.$$ (79)

$t$ is a PARAMETER, chosen so that $t=t_k$, such that

$$\lim_{k\to \infty} y(b, t_k)=y(b)=\beta$$

$$1^{st} \left\{\begin{array}{ll} y''=f(x,y,y') & x_0\le x\le b\\ y(x_0)=\alpha\\ y(x_0)=t_0 \end{array}\right.$$

If $y(b,t_0)$ not close enough to $\beta$, we choose another ``elevation'' $t_1$, and check to see if close enough. If not, choose the next ``elevation'' $t_2, \cdots$ until our ``shots'' get close to $\beta$. The situation is depicted in Figure (14), which clearly shows why the method bears its name.

Figure 14: Nonlinear shooting method

How to choose $t_k$? If $y(x,t)$ is approx solution to (79) (The IVP) then we need to determine $t$ such that

$$y(b,t)-\beta=0$$

A nonlinear equation that can be solved using an efficient root-finding method For example, using secant:

$$t_k=t_{k-1}-\frac{(y(b,t_{k-1})-\beta)(t_{k-1}-t_{k-2})}{y(b,t_{k-1})-y(b,t_{k-2})} \qquad k=2,3, \ldots$$

A better and more elegant method uses the Newton method integrated into the IVP sequence. See Burden and Faires for algorithm page 587.

The shooting method, when it works, is usually quite fast. It is easy to implement. Its good qualities are offset by possible instabilities. An alternative method, such as finite difference method, to be shown next.