I have to numerically solve a nonlinear partial integro-differential equation. This is my equation,
$$\frac{\partial y(x,t)}{\partial t}=\int_{-\infty}^\infty K_0(|x-u|) \frac{\partial^2 y(u,t)}{\partial u^2}\mathrm du-\sin\,y(x,t)$$
The kernel $K$ is the modified Bessel function of the 2nd kind which has a logarithmic singularity at zero and decays as exponential (logarithmic singularities are integrable) and answer $y$ is a function of $x,t$.
Notice, the derivative inside the integral is a second derivation in spacial component and the equation has a non-linearity of $sin(y(x,t))$. Does this equation belong to any special class of integral equations? (Fredholm, Volterra ...) Please experts who have some experience with numerical recipes dealing with integral equations reply.