I am having some trouble with using the shooting method : I am given a system of ode's representing initial value problem, and I know I want one of the unknowns (=u) to vanish at infinity (which I took for simplicity to be 30).

The problem is that when I am using the shooting method (I have a one-dimensional shooting. i.e.- only one of the unknowns is undetermined in (0) ), I get an oscillatory solution (the oscillations have a growing amplitude) that indeed satisfy u(30)=0, but it doesn't approach 0 uniformly (rather, it is $10^{23}$ right before 30) . The ODE becomes unstable in the direction I am shooting, so that the dominant mode completely contaminates the search for the subdominant boundary condition

Is there any suitable method for such a case ? (i.e.- I am looking for a variant of the shooting method that is able to look for a solution at infinity which is not oscillatory. rather, it should approach 0 a smooth way)

Hope I made myself clear enough.

Thanks in advance

  • $\begingroup$ A quick note is that scicomp.SE might be more helpful, if the question doesn't get any attention here in a reasonable amount of time. $\endgroup$ – Igor Khavkine Jan 1 '15 at 19:12
  • $\begingroup$ I'm not quite sure about the question itself. It's unreasonable to expect your solution to vanish in an entire neighborhood of your "infinity". For a single linear scalar ODE, such a strong restriction would be equivalent to the solution vanishing everywhere, not just at infinity. Is the problem that the ODE becomes unstable in the direction that you are shooting, so that the dominant mode completely contaminates the search for the subdominant boundary condition? If that is so, perhaps shooting in the other direction would help. But it's hard to say more without more details. $\endgroup$ – Igor Khavkine Jan 1 '15 at 19:20
  • $\begingroup$ @IgorKhavkine : thanks a lot for your suggestion for a more appropriate forum and for your answer. I edit my first post , and included your explanation in it. Hope it is ok. Indeed, the problem is that the ODE is unstable in the vicinity of 30, and the oscillations amplitude becomes very large (it is expected to be bounded, but increases to $10^23$ and more) . I have tried shooting in the other direction, but still, I prefer to use the shooting in this direction (I know the values of all of the relevant derivaties in 0, which I do not know in 30) $\endgroup$ – NumericsWannabe Jan 2 '15 at 7:10
  • $\begingroup$ so I would prefer to use another numerical method that is able to "ignore" oscillatory solutions, with growing amplitude, and to only take into account the stable solution. Indeed, I did not mean the solution would vanish in an entire vicinity of 30, but that it would approach "0" in a stable, smooth way, and not in an unstable one, as you suggested. Thanks a lot ! $\endgroup$ – NumericsWannabe Jan 2 '15 at 7:11
  • $\begingroup$ I'm not an expert, so I don't really have anything specific to recommend. My only remaining suggestion is that it might be possible to use an asymptotic expansion to obtain the needed initial conditions for shooting in the other direction (from "infinity" to zero). $\endgroup$ – Igor Khavkine Jan 2 '15 at 17:55

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