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

$$\frac{\partial y(x,t)}{\partial t}=\int_{-1/2}^{1/2} \frac{\pi\cos u}{\sin\pi u-\sin\pi x} \frac{\partial y(u,t)}{\partial u}\mathrm du-\sin\,y(x,t)$$

The kernel has a singularity at $x=u$ and vanishes at the boundaries ($u=-1/2,1/2$). Answer $y$ is a function of $x,t$. At the moment I am not worried about the initial conditions.

Notice, the derivative inside the integral is a first derivation in spacial component and the equation has a non-linearity of $\sin(y(x,t))$. No boundary condition is needed to solve this equation (It's all been taken care of by the kerenl).

What would be the possible numerical recipes to solve this equation?

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    $\begingroup$ So far, your question is not about anything numerical. $\endgroup$ – David Ketcheson Mar 8 '15 at 4:24

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