I try to solve an integro-differential equation with a convolution term using this code. Here i used a periodic condition.

    L=20;

    NDSolve[{D[u[x, t], t] + u[x, t]*D[u[x, t], x] + D[u[x, t], {x, 2}] + 
    D[u[x, t], {x, 4}] + 1/\[Pi]*D[Convolve[u[xp, t], 1/xp, xp, x,
    PrincipalValue -> True], {x,3}] == 0, u[-L, t] == u[L, t],
    u[x, 0] == 1/10*Cos[x]}, u, {x, -L, L}, {t, 0, 2}]

which gives me
>NDSolve::delpde: Delay partial differential equations are not currently supported by NDSolve.

I also tried `Method -> "MethodOfLines"`, but it still doesn't work.

My problem may closely relate to [this][1], which is more like a wave equation in the sense of $u_{tt}$. But my pde has 1st-derivative in $t$. So I post my problem here for a more general solution. Thank you for any suggestion.


  [1]: https://mathematica.stackexchange.com/questions/107538/solving-pde-involving-hilbert-transform-numerically