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I have to numerically solve a nonlinear partial integro-differential equation using Mathematica. This is my equation,

$$\frac{\partial y(x,t)}{\partial t}=\int_{-\infty}^\infty K_0(|x-u|) \frac{\partial^2 y(u,t)}{\partial u^2}\mathrm du-\sin\,y(x,t)$$

D[y[x, t], t] == Integrate[BesselK[0, u] D[y[u, t], {u, 2}], {u, -∞, ∞}] - Sin[y[x, t]]

$K$ is the modified Bessel function of the 2nd kind and answer $y$ is a function of $x,t$.

what I want to do is to first try to solve it with some built-in functionality with MATLAB or MATHEMATICA if possible. If not, what is your recommendation to go around this IDE. any help would be really appreciated.

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closed as off-topic by Jens, m_goldberg, MarcoB, ilian, J. M. Aug 13 '15 at 22:36

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I tried to $\TeX$ up your equation, but it isn't clear which partial derivatives you're taking. Where did the equation come from? – J. M. Aug 17 '12 at 4:05
What I would try to do is to first remove the second derivative under the integral by integrating by parts twice, and then integrate your equation over time numerically iteratively, discretizing the time variable and computing the value of y(t+dt) from y(t) using your equation (its r.h.s) with y(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 13:03
Guys,first of all I should say that K(x,u) is first modified Bessel function and u is just a variable under the Integral.secondly I haven't tried your answers so far but I think Leonid's is understandable.@ Leonid:Does any solver exist in mathematica for this? – Ahmad Sheikhzada Aug 20 '12 at 11:16
Maybe this Handbook of integral equations book could provide some clue. – Silvia Aug 23 '12 at 4:34
I'm voting to close this question as off-topic because it asks mainly for the mathematical approach and isn't pinpointing a specific Mathematica issue. – Jens Aug 13 '15 at 20:55