# numerical analysis of partial integro-differential equation

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.

• I'm no expert on this class of equations, but I would ask you whether you expect in advance that the solution is stable. – Ian Mar 7 '15 at 0:52
• Yes, This equation describes a physical phenomenon which has been simplified for the sake of argument. – Ahmad Sheikhzada Mar 7 '15 at 0:56
• Do you have a (or some) particular numerical method in mind? What are the boundary conditions? (I assume this is an initial value problem and $y$ decays to zero at infinity). How do you need the solution parameterized? – Victor Liu Mar 7 '15 at 0:56
• Well, this is an integral equation which all boundary conditions are represented using that kernel, so to be solved it does not need any boundary condition, but yes! the solution would change from zero to 2$Pi$ in (-$infinty$ $infinity$) – Ahmad Sheikhzada Mar 7 '15 at 0:59
• I honestly wrote down a code using method of lines which is simply defining a mesh of spacial points over a bounded space which I am more interested (say N points), thus I will get N coupled ODEs which can be solved by ODE solvers in MATLAB, but I do not trust this and it does take a long time due to the singularity. I will give you more info if you ask. – Ahmad Sheikhzada Mar 7 '15 at 1:09