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 ____________________________________________________________________