Using Mathematica to locate a PDE singularity

• To: mathgroup at smc.vnet.net
• Subject: [mg66662] Using Mathematica to locate a PDE singularity
• From: "Alan" <info at optioncity.REMOVETHIS.net>
• Date: Fri, 26 May 2006 04:17:27 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```This is a very long-shot question, but I will give it a try anyway.

I have  a diffusion process with a generator
A f (x) = a(x) f_xx + b(x) f_x
where the subscripts indicate differentiation

Now introduce the complex parameter c, independent of x.
I am interested in the behavior (vs. c) of solutions to the PDE problem

dg(x,t;c)/dt = A g - c x g  on R+ = (0,Infinity),  with  g(x,0;c) = 1.

Assume the precise nature of the boundary at x=0 is irrelevant
but that x=Infinity is a natural boundary, unreachable by the diffusion
in finite time.

For fixed t, I suspect there is a singularity on the negative real c-axis.
Call it c*(t). I don't know its nature, except that I believe c*(t1) <
c*(t2) < 0.
for t1 < t2, so that the problem is well-posed on [0,T] for c > c*(T).

I truncate and solve this PDE problem with NDSolve on
(0, xmax) with a reflecting boundary condition at xmax. This works fine,
at least up to xmax not too large and c not too negative, at which point
the solver will choke.

Assume that as xmax-> Infinity, this approximate solution converges to the
one
on R+. The problem is that, since xmax < Infinity, the approximate solution
g(x,t;c) is
now an entire function of c (analytic everywhere).

Any suggestions on how to use Mathematica to locate my putative
singular point c*(t)? So far I have tried plotting 1/Abs[Log[g(x,t;c)]] vs
c,
but this does not seem to be leading anywhere helpful.

Thanks for reading this far!
alan

```

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