[SciPy-user] solving an ode with boundary conditions

[Max Wainwright]

On Jun 23, 2009, at 6:57 PM, Rob Clewley wrote:

> On Tue, Jun 23, 2009 at 8:37 PM, Max Wainwright<cwainwri at ucsc.edu>  
> wrote:
>> I have a 1D second-order differential equation that I'm trying to
>> solve with the following boundary conditions: dy/dt = 0 at r = 0  
>> and y
>> (t) = 0 at t = infinity.  I can solve this by doing guess and check
>> for the initial value of y at t = 0.  If I guess too high, then y
>> goes to negative infinity at t = infinity.  If I guess too low, y
>> never reaches zero and instead oscillates about some minimum.
>> Therefore, to check if I've overshot or undershot the solution all I
>> need to do is stop the integration once either y or -dy/dt goes
>> negative.  Is there any way to do this with scipy?  I also tried
>> looking at PyDSTool, but I had a hard time finding what I need.  I
>> don't in principle know the time-scale of the solution (I'll be doing
>> this for lots of different parameters), so I'd like to avoid using a
>> fixed timestep.
>
> It's pretty easy to set up a standard minimizer from scipy to use an
> adaptive time step solver of your choice to implement the "shooting
> method" for this boundary value problem (BVP), which is simple enough
> to permit this method of solution. You just need to measure your
> over-/undershoot numerically in such a way that it provides correct
> feedback to the optimizer. PyDSTool would make catching the event of y
> or -dy/dt going negative very simple, but it does not have a BVP
> solver itself either. There are plenty of tutorial and demo scripts
> about that online. This would be hard with scipy's solvers. If you set
> up an attempt and need more help you can post it here for more
> feedback.
>
> Scipy does not have a BVP built in either. You can also try this
> recent package:
> http://www.elisanet.fi/ptvirtan/software/bvp/index.html but that might
> be like using a sledgehammer to crack a nut, as we say.
>
> -Rob
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Could you kindly point me to an example PyDSTool script that does  
some sort of variable tracking (ie, checking when y goes negative)?   
I'm completely new to the package, and I can't find it in the wiki.

I don't think I can use the standard odeint or ode from scipy,  
because I don't know the point at which the over or undershooting  
will become obvious.  It could be at r = .1, or it could be at r =  
100.  I suppose I could dynamically change the integration range, but  
this seems kind of kludgy.

Thanks again.

-Max