I have to numerically solve a nonlinear partial integro-differential equation using Mathematica. This is my equation,
$$\frac{\partial y(x,t)}{\partial t}=\int_{-\infty}^\infty K_0(|x-u|) \frac{\partial^2 y(u,t)}{\partial u^2}\mathrm du+\sin\,y(x,t)$$
$K$ is the modified Bessel function of the 2nd kind and answer $y$ is a function of $x,t$.
This is the basic code I have written so far but it is far away from completeness.
T = 2 \[Pi]; L = 10;
kernel[x_, y_] := BesselK[0, Abs[x - y]]
intkern[x_?NumericQ, u_] :=
NIntegrate[kernel[x, y]*D[u[t, y], y, y], {y, -Infinity, Infinity}]
u[0, x] = 0;
u[t, 0] = 0;
u[t, L] = 2 \[Pi];
soln1 = Evaluate[
First[u[t, x] /.
NDSolve[{D[u[t, x], t] ==
intkern[x, u[t, 0], u[t, L], y] + Sin[u[t, x]] + 2,
u[t, 0] == 0}, u, {t, 0, T}, {x, 0, L}]]];
Plot3D[u[t, x], {t, 0, T}, {x, 0, L}]
y(t+dt)fromy(t)using your equation (its r.h.s) withy(t,x)obtained at the previous step. I actually used similar methods with success in the past, also in Mathematica. – Leonid Shifrin Aug 17 '12 at 13:03yis a function of x and t. Why does the integral show it to be a function ofu? Souis a function of x and perhaps t? – drN Aug 19 '12 at 23:44