       Odd behaviour of solution of PDE

• To: mathgroup at smc.vnet.net
• Subject: [mg116597] Odd behaviour of solution of PDE
• From: Alan Ford <fabio.sattin at igi.cnr.it>
• Date: Mon, 21 Feb 2011 05:34:21 -0500 (EST)

```Dear all,

I encountered an odd behaviour with this PDE:

dy/dt = d(D(x) dy/dx - U(x) y) + S(x,t)

It is a standard diffusion-convection equation with time-and space-
dependent source term.
I wrote it in cartesian coordinate, but actually in my problem it is
written in cylindrical coordinates,
thus x is the radius:   eps < x < 1.  eps is a very small number, set
different to zero to avoid trouble
with the singular point at the origin.

The PDE is supplemented with initial condition y = 0 at t = 0, and two
boundary conditions:
one is dy/dx = 0 at x = eps, and the other is a time-dependent BC at x
= 1, y(x=1) = y_BC(t).

The odd thing is this: I ask Mathematica (v. 7) to solve for y between
eps < x < x0
with x0 < 1, NO MATTER how smaller than unity (say, x0 = 0.999).
Everything works
apparently fine, i.e, the solution reaches asymptotically a stable
state.
Now, I set the outer bound of integration to 1, i.e, place it at the
location where the boundary condition
is fixed. Then, the solution diverges in time.
Notice that there nothing special in 1, and I made checks with other
values: whenever I
set x0 = x_BC, the solution looks wrong. On the other hand, as soon as
I make
the difference x_0 - x_BC as small as I wish (but non-zero, say
10^(-5)), then everything is OK.

Anybody has guesses about why this strange behaviour ?

Thanks for any help

Fabio

```

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