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Re: Q: Combining NDSolve with FindRoot
*To*: mathgroup at smc.vnet.net
*Subject*: [mg13382] Re: Q: Combining NDSolve with FindRoot
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Thu, 23 Jul 1998 03:32:30 -0400
*Organization*: University of Western Australia
*References*: <6okkoi$1lf@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
[Contact the author to obtain the notebook or go to http://smc.vnet.net/paul.nb -
Moderator]
Anil Trivedi wrote:
> Trying to learn mathematica, I thought I would verify the following.
> The "harmonic oscillator" equation:
>
> y''[x] + (2e-x^2) * y[x] =0,
> y[0]=1, y'[0]=0
>
> has solutions which vanish for large x only if e is one of the
> eigenvalues e= 0.5, 2.5, 4.5, 6.5, etc.. How can I generate this
> series, or the exact function e[n] = 2n+1/2 where n=0,1,2,..?
See the attached Notebook (taken from an exam question from my
computational physics course here at UWA) which addresses a similar
question.
> Focussing on the first eigenvalue e = 0.5, let us try to (i) solve the
> equation with NDSolve,
BTW, here is one way of doing this (assuming that you already know the
eigenvalue):
NDSolve[{y''[x] + (1 - x^2)*y[x] == 0, y[0] == 1, y'[0] == 0},
y[x], {x, -5, 5}];
Plot and compare with the exact solution:
Plot[Evaluate[{HermiteH[0, x]/E^(x^2/2), y[x] /. First[%]}],
{x, -5, 5}, PlotStyle -> {Hue[1/3], Hue[1]}];
>(ii) evaluate the soln at some large x = L,
> (iii) call the resulting function z[e], and (iv) use FindRoot to solve
> z[e]=0, with a good intitial guess like 0.45. :)
Basically, I think that you would need to use a series solution method,
after factoring off the appropriate asymptotic form, to compute the
eigenvalue in this way. Alternatively, the Notebook demonstrates a
general matrix method for the approximate determination of the
eigenvalues.
> 3. Assuming I can do this for one eigenvalue, what is the best
> "mathematica way" of iterating the procedure to obtain the first N
> eigenvalues? (I doubt it is Do loop, but I don't know what it is.)
The matrix method with an n x n matrix yields the first n eigenvalues.
Cheers,
Paul
____________________________________________________________________
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia Nedlands WA 6907
mailto:paul at physics.uwa.edu.au AUSTRALIA
http://www.pd.uwa.edu.au/~paul
God IS a weakly left-handed dice player
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