I am trying to understand the shooting method and how to use it. I need it for some more complicated differential equations with boundary conditions at infinity, but I wanted to start with simple things.
I found the following example of BVP in Wiki
$ $ w”(t)=\frac{3}{2}w^2(t)$ $
with $ w(0)=4$ and $ w(1)=0$
The IVP that corresponds to the above
$ $ w”(t)=\frac{3}{2}w^2(t)$ $
with $ w(0)=4$ and $ w'(0)=s$
My implementation in Mathematica.
sols = Map[ NDSolve[{D[w[t], {t, 2}] == 3/2 w[t]^2, w[0] == 4, w[1] == 1}, w, t, Method -> {"Shooting", "StartingInitialConditions" -> {w[0] == 4, w'[0] == #}}] &, {-36, -8, -4, -3, -2, -1, 2, 3}] {{{w -> InterpolatingFunction[{{0., 1.}}, <>]}}, {{w -> InterpolatingFunction[{{0., 1.}}, <>]}}, {{w -> InterpolatingFunction[{{0., 1.}}, <>]}}, {{w -> InterpolatingFunction[{{0., 1.}}, <>]}}, {{w -> InterpolatingFunction[{{0., 1.}}, <>]}}, {{w -> InterpolatingFunction[{{0., 1.}}, <>]}}, {{w -> InterpolatingFunction[{{0., 1.}}, <>]}}, {{w -> InterpolatingFunction[{{0., 1.}}, <>]}}}
So, I get solutions without errors, but when I plot them I only get two discrete lines.
Plot[Evaluate[w[t] /. sols], {t, 0, 10}]
In Wiki they mention the following:
“was solved for $ s = −1, −2, −3, …, −100$ , and $ F(s) = w(1;s) − 1$ plotted in the first figure. Inspecting the plot of F, we see that there are roots near $ −8$ and $ −36$ . Some trajectories of w(t;s) are shown in the second figure.”
Because of that, I expected that the plots of the solutions to the D.E should be very different since not all the values for $ s$ are roots. And this is my first question. Am I doing something wrong or did I understand something wrong?
The second one: What do they actually mean by $ w(1;s)$ in order to determine which values of $ s$ are the right ones? I don’t know if I should ask this here or in the MathStack.
The relevant wiki link
https://en.wikipedia.org/wiki/Shooting_method