13
$\begingroup$

I am trying to simulate a stochastic differential equation in time and space, but I'm unsure if this can be done in Mathematica. The sde that I would like to study is:

$$ dN[x,t]=N[x,t](1-N[x,t])dt+\sqrt{N[x,t]}dw+\partial^2_xN[x,t]dt $$

With an exponential decreasing (in space) initial condition N[x,0]=Exp[-|x|]. Is this possible to simulate with ItoProcess?

Thank you, Best, Andrea

$\endgroup$
3
  • 1
    $\begingroup$ it can be done, for example, by discretising spatially. $\endgroup$
    – acl
    Apr 12, 2013 at 10:22
  • $\begingroup$ Sorry but I don't see easily how, in particular I feel I would have no control on step sizes in time $\endgroup$
    – Andrea
    Apr 13, 2013 at 8:56
  • 3
    $\begingroup$ discretize spatially, this solving a set of coupled SDEs in time. like the method of lines for PDEs. $\endgroup$
    – acl
    Apr 13, 2013 at 9:15

1 Answer 1

7
$\begingroup$

Here's a solution following @acl's suggestion of discretizing in space. I added a diffusion coefficient d and used reflecting boundary conditions.

l = 1.0; (* length of domain *)
nx = 101; (* number of spatial cells *)
dx = l/(nx - 1); (* size of each cell *)
d = 0.1; (* diffusion coefficient *)

eqns = Join[
  {\[DifferentialD]n[1][t] == (n[1][t] (1 - n[1][t])
   + d (-n[1][t] + n[2][t])/dx^2) \[DifferentialD]t
   + Sqrt[n[1][t]] \[DifferentialD]w[1][t]},
   Table[
     \[DifferentialD]n[i][t] == (n[i][t] (1 - n[i][t])
     + d (n[i - 1][t] - 2 n[i][t] + n[i + 1][t])/dx^2) \[DifferentialD]t
     + Sqrt[n[i][t]] \[DifferentialD]w[i][t]
   , {i, 2, nx - 1}],
   {\[DifferentialD]n[nx][t] == (n[nx][t] (1 - n[nx][t])
   + d (n[nx - 1][t] - n[nx][t])/dx^2) \[DifferentialD]t
   + Sqrt[n[nx][t]] \[DifferentialD]w[nx][t]}
];

noise = Table[w[i] \[Distributed] WienerProcess[0, 0.5], {i, nx}];

unks1 = Table[n[i][t], {i, nx}];
unks2 = Table[n[i], {i, nx}];

ics = Table[Exp[-(i - 1) dx], {i, nx}];

tmax = 1;
dt = 0.0001;
sol = RandomFunction[ItoProcess[
  eqns, unks1, {unks2, ics}, t, noise], {0, tmax, dt}]

This give an Function::flpar message, but still runs.

ListLinePlot[sol, PlotRange -> {All, All}]

enter image description here

Here's an animation of n[x]:

Export["space.gif", 
 Table[ListLinePlot[sol["SliceData", t], PlotRange -> {0, 1}], {t, 
   0.0, 1.0, 0.01}]]

enter image description here

$\endgroup$
1
  • $\begingroup$ can you tell me if it was N[x,y,t], then how would someone define w and n in the above program? $\endgroup$ Nov 3, 2022 at 14:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.