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Re: Modified shooting method

• To: mathgroup at smc.vnet.net
• Subject: [mg30511] Re: Modified shooting method
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Fri, 24 Aug 2001 20:58:07 -0400 (EDT)
• Organization: Universitaet Leipzig
• References: <9m522e$qdc$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Hi,

eigenvalue problems work not very well with simple shooting
methods.
The problem ist, that for a fixed value w[i] you have 
 - no solution if w[i] is not a eigenvalue
 - an infinite number of solutions if w[i] *is* an eigenvalue

The bes solution is to use a double or multiple
shooting method.  You should have a look at

http://www.netlib.org/ode/index.html

file    mus.doc  mus.doc plus dependencies
by      Mattheij and Staarink
for     ordinary differential equation boundary-value problem solver
alg     multiple shooting
ref     http://www.win.tue.nl/math/an/noframe/ftp/index.html
,       whence you may also fine DEQNS
prec    double
gams    I1b1 I1b2



An better way may be to use a matrix approach and solve 
the matrix eigenvalue problem. The simplest idea is to
us finite differences 
y''[x] -> (y[i-1]-2 y[i]+y[i+1])/h^2

for the inner points, and solve
the problem

(1-BesselI[0,x]/BesselI[0,rap])-1-3/4/r^2-BesselI[1,r]/Bessel[0,rap]/r)*(y[i-1]-2
y[i]+y[i+1])/h^2+w^2*y[i]

Or to expand the solution in a orthogonal basis you like,
and solve the eigenvalueproblem
Hope that helps
  Jens

Christophe Le Poncin-Lafitte wrote:
> 
> Hello,
> 
> Sorry for my english, which is poor.
> I'm working about a modelization of cyclonic and anticyclonic structures in
> geophysic ; so this is a study of the nonlinear adjustment of a density
> front.
> During our experiment, we observe the adjustment of a cylindrical structure,
> which becomes a little stable lens (during 40s), before the development of
> instabilities. But during the adjustment, we observe the development of
> inertial-gravity waves.
> Now we search to modelize our system.
> Our goal is to quantify the energy of this waves, to evaluate the importance
> of these for the comprehension of the adjustment first, then to understand
> the relative stability of the system, and finally to find why we observe
> instabilities, which destroy the system.
> 
> I have a little problem to solve : a boundary value problem.
> My ODE is quite difficult :
> 
> y"[x]+q[w,x]*y[x]==0
> 
> where
> q[w,x]=w^2/(1-BesselI[0,x]/BesselI[0,rap])-1-3/4/r^2-BesselI[1,r]/Bessel[0,r
ap]/r


> 
> rap is a parameter, which is a constant. w is the eigenmodes of
> inertial-gravity waves.
> 
> My boundary conditions are  : y[0]=0 and y[rap]=0
> I take for initial condition on y' : y'[0]=1.
> 
> In fact, I have to done my shooting on w, to isolate the different
> eigenmodes of inertial-gravity waves. So, this is a modified shooting
> method.
> I want to know if anybody has already treat this type of problem ?
> 
> Regards,
> Christophe