ODE problem

*To*: mathgroup at yoda.physics.unc.edu*Subject*: ODE problem*From*: Silvio Levy <levy at math.berkeley.edu>*Date*: Wed, 10 Nov 93 13:17:57 -0800

I was asked by Al Edelson (aedelson at galileo.ucdavis.edu ) the following question: > I told you on the phone I am looking for a program that will plot > bifurcation curves for semilinear elliptic equations in Rn. It is > enought to assume radial symmetry , so that I can consider only an ode. > The problem is then > > (1) u''(r) + [(n-1)/r] u'(r) =(lambda)f(r)g(u), > f(r)>0, 0<r< infinity, 0<lambda, > > f is a positive function which goes to zero at infinity fast enough to > ensure that there is a positive, decreasing solution asymptotically > equivalent to r^(alpha). Typically g(u) is a power, g(u)=u^(gamma). In > many cases alpha is (2-n), and the solution is unique. There are other > cases where alpha = (2-n)/2. I want to find the norm of this decreasing > solution corresponding to the parameter value lambda; the norm is > defined by ||u|| = limiting value of r^(-alpha)u(r); then plot lambda > against ||u||. > > > The steps are: > 1. Input numbers lambda0, u0, stepsize, u0, lambdaminimum, alpha. > (I am assuming that I will have to select some initial parameters.) > 2. Set lambda = lambda0. > 3. Set initial conditions u(0)=u0, u'(0)=0. > 4. Solve numerically for u(r), and calculate r^(alpha) u(r). > 5. Decrease u(0) by stepsize until r^(alpha)u(r) is asymptotically > constant. In fact,for large u(0), we will see that u(r) has a positive > zero, so we decrease u(0) by stepsize and we will find a critical value > of u(0) for which r^(alpha)u(r) is asymptotically constant. > 6. Find the limiting value of r^(alpha)u(r)=||u||. > 7. Construct an array containing (lambda, ||u||). > 8. Decrease lambda by stepsize and repeat until lambda < > lambdaminimum. (In some cases the solution would get large as lambda > gets small, or as lambda approaches some positive lambda0, so I have to > detect that situation.) > 9. Plot lambda vrs ||u||. > > I have left out all the details, since I don't know how to do it. For > example, how do we decide that r^(alpha) u(r) is "asymptotically > constant"? Also there will be cases where the solution is non unique, > and we would like to look for more than one solution. But I hope this > conveys the idea of what I am trying to accomplish. Klaus Schmidt and > Renata Schaaf have a program for Dirichlet problems on [0,1], but as > far as I know nobody has done it for unbounded domains. If you > know of any programs of this sort I'd be very > interested. (I use a sun sparc 2.)