0
$\begingroup$

When considering the time evolution of a distribution over a state variable $x$, one of the cases that seems fundamental is when knowledge about $x$ begins uniform. However, modelling the process as drift-diffusion, makes absorbing boundaries inconsistent.

Take the evolution of the distribution $P(x,t)$ to be governed by:

$\frac{\partial P}{\partial t} = -\mu \frac{\partial P}{\partial x} + \frac{\sigma^2}{2} \frac{\partial^2 P}{\partial x^2} $

If we consider absorbing boundaries at $(a,b)$ then the boundary conditions are:

$P(a,t) = 0$ and $P(b,t) = 0$

However, the initial condition assuming a Uniform distribution is:

$P(x,0) = \frac{1}{b-a} $

Together, this implies an inconsistency.

$P(a,t) = P(b,t) = 0 = \frac{1}{b-a} $

How can this issue be thought about in such a way that it can be resolve?

One pathway I have considered is that the absorbing boundaries are somehow $\mathrm{d}x$ outside the state-space.

Ultimately, this is part of a larger project that will require numerical solution. Here, I provide Mathematica code which exemplifies the problem, it provides the error that boundaries are inconsistent.

Yet, Mathematica still provides a solution, what is it doing in this case, and will the solution it provides still be fairly accurate?

Any help is greatly appreciated!

ClearAll["Global`*"]
a = 0;
b = 5;
\[Mu] = .5;
\[Sigma] = .5;

Tmax = 10;

FPEq = D[P[x, t], 
    t] == -\[Mu]   D[P[x, t], {x, 1}] + \[Sigma]^2/
     2 D[P[x, t], {x, 2}];
iC = P[x, 0] == 
   PDF[TruncatedDistribution[{a, b}, UniformDistribution[{a, b}]], 
    x];
bC = {P[a, t] == 0, P[b, t] == 0}; 


sol = NDSolve[{FPEq, iC, bC}, P, {x, a, b}, {t, 0, Tmax}];


Psol[x_, t_] := Evaluate[P[x, t] /. First[sol]];

DensityPlot[Psol[x, t], {t, 0, Tmax}, {x, a, b}, AspectRatio -> 1/2, 
 PlotTheme -> "Scientific"]
$\endgroup$
7
  • 1
    $\begingroup$ It is weird indeed. If you evaluate the solution, you will see that $P(b, \cdot)$ is not $0$ for example. So Mathematica seems to ignore the spatial boundary conditions. Maybe you should use Neumann boundary conditions $\endgroup$
    – Hyperbolic PDE friend
    Commented May 6 at 13:32
  • 1
    $\begingroup$ Thanks for you input @Hyperbolic PDE friend. Unfortunately, absorbing boundaries are a necessary part of my set up, I don't know I can achieve that with Neumann boundaries $\endgroup$
    – CRTmonitor
    Commented May 6 at 14:00
  • 1
    $\begingroup$ In that case, maybe reevaluate your initial condition. Otherwise you could also go for a discontinuous boundary condition $\endgroup$
    – Hyperbolic PDE friend
    Commented May 6 at 14:04
  • 1
    $\begingroup$ Thanks again @HyperbolicPDEfriend, if I use piecewise initial conditions, as follows, I get a different (although similar solution). What I'd be concerned about now is that I'm not defining a probability distribution, by considering the open rather than closed interval. iC = P[x, 0] == Piecewise[{{1/(b - a), a < x < b}}, 0]; One thing I'm considering is whether the diffusion constant can be written in terms of the boundary dimension, might be another way into the problem. $\endgroup$
    – CRTmonitor
    Commented May 6 at 14:21
  • 1
    $\begingroup$ About the diffusion constant I can't say a lot unfortunately. However, about your choice of $P(x, 0)$ I would say that it is not that bad, because the change only occurs on a set of measure zero..... $\endgroup$
    – Hyperbolic PDE friend
    Commented May 6 at 14:42

0

Reset to default

You must log in to answer this question.