Solving partial integro-differential equation

Question

I want to solve a partial integro-differential equation of this form:

using NDSolve, something like

NDSolve[{D[u[t, x], t] == D[u[t, x], x, x] - D[u[t, x]*int[w (s)*u[t, x + s]], {s, -L, L}],x] + u[t,x] * (1-u[t,x]), u[0, x] == u0, u[t, -L] == u[t, L],u, {t, 0, tend}, {x, 0, 2 xend}],

but can't find how to interpolate such integral with Simpson's rule or something similar. Any idea?

Answer 1

I use the code from my answer on Solving partial differential equation involving Hilbert transform

We put $\mu =0,\sigma^2 =1/2$,L=Infinity,n[0,x]==Cos[x]

n = Sum[f[m][t] Exp[I m x], {m, -Infinity, Infinity}]

Then the integral is calculated as

 Integrate[
 Exp[-s^2] Exp[I m s]/Sqrt[Pi], {s, -Infinity, Infinity}]

(*Out[]= E^(-(m^2/4))*)

Now we can make a system of equations

nn = 137; tm = 1.; eq = 
 Table[-f[m]'[t] - m^2 f[m][t] - 
    I m Sum[If[Abs[m - k] <= nn, f[m - k][t], 0] Exp[-k^2/4] f[k][
        t], {k, -nn, nn}] + f[m][t] - 
    Sum[If[Abs[m - k] <= nn, f[m - k][t], 0] f[k][t], {k, -nn, nn}] ==
    0, {m, -nn, nn}];
ic = Table[
   f[m][0] == (KroneckerDelta[m, 1] + KroneckerDelta[m, -1])/
     2, {m, -nn, nn}];
var = Table[f[i], {i, -nn, nn}];
sol = NDSolveValue[{eq, ic}, var, {t, 0, tm}];

Data visualization

{Plot[Evaluate[
   Table[Re[
     Sum[sol[[m + 1]][t] Exp[I (-nn + m) x], {m, 0, 2*nn}]], {t, 0, 
     tm, .2}]], {x, 0, 2*Pi}, Mesh -> None, ColorFunction -> Blue, 
  AxesLabel -> Automatic, PlotLegends -> Automatic, PlotRange -> All],
  Plot3D[Re[
   Sum[sol[[m + 1]][t] Exp[I (-nn + m) x], {m, 0, 2*nn}]], {t, 0., 
   tm}, {x, 0, 2*Pi}, Mesh -> None, ColorFunction -> Hue, 
  AxesLabel -> Automatic, PlotRange -> All]}