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.)

```

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