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