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

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