The Shooting Method

Linear Case:

Take

$$Y''=p(x)Y'+q(x)r \quad x_0\le x\le b \quad Y(x_0)=\alpha, Y'(x_0)=0$$ (75)

and

$$Y''=p(x)Y'+q(x)Y\quad x_0\le x\le b\quad Y(x_0)=0, Y'(x_0)=1.$$ (76)

If $p, q, r$ continuous and $q > 0$ on $[x_0, b]$ then the Lipschitz condition exists for cast as a system $\Rightarrow$ both (75) and (76) have unique solutions.

Take $Y_1(x)$ solution of (75) and $Y_2(x)$ solution of (76)

$$\Rightarrow Y(x)=Y_1(x)+\frac{\beta-Y_1(b)}{Y_2(b)}Y_2(x),$$ (77)

(provided $Y_2(b)\ne 0.$ can be checked to be unique solution of

$$BVP\equiv\left\{\begin{array}{ll} Y''=p(x)Y'+q(x)Y+r(x) & x_0\le x\le b Y(x_0)=\alpha Y(b)=\beta \end{array}\right.$$ (78)

Remark: Note that if $Y_2$ is solution of $Y''=p(x)Y'+q(x)Y$ and $Y_2(x_0)=Y_2(b)=0\Rightarrow Y_2=0$.

Summary:

So the shooting-method strategy amounts to the following: Replace (78) by 2 IVP (75) and (76). Use appropriate method to solve (75) and (76) and piece solution as per (77). Figure (13) shows graphically the construction of the solution $Y(x)$ in terms of $Y_1(x)$ and $Y_2(x)$.

Figure 13: Graphical construction of the solution.

ALGORIHM (from Burden and Faires $p$ 582)

1. Set $h=(b-x_0)/N$
   $u_{1,0}=\alpha\\ u_{2,0}=\alpha\\ v_{1,0}=0\\ v_{2,0}=1$

2. for $i=0\ldots N-1$
   
   

   $$\left[\begin{array}{l} \mbox {Use ERK4 (or some other suitable IV... ...solve for}\\ u_{1, i+1}, u_{2, i+1}, v_{1, i+1}, v_{2, i+1} \end{array}\right.$$

   
   

$$w_{1,0} = \alpha$$

$$w_{2,0} = \frac{\beta-u_{1,N}}{v_{1,N}}$$

$$(x_0; w_{1,0} w_{2,0})$$

$$w_{1,0} \mbox { is an approximation to } Y(x_0) \mbox { and } w_{2,0} \mbox{ an approximation to } Y'(x_0)$$

$$i = 1, \cdots N$$

$$x = x_0+ih$$

$$w_1 = u_{1,i}+w_{2,0}v_{1,i}$$

$$w_2 = u_{2,i}+w_{2,0}v_{2,i}$$

$$\mbox {output }(x, w1, w2)$$

$$w_2 Y'(x_i)$$

$$w_1 Y(x_i)$$

END

$\Box$