Solving an integro-differential equation with Mathematica

I try to solve a nonlinear integro-differential equation with this code. Here i used a periodic condition.

L=10; tmax = 2;

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/(2 L)*NIntegrate[D[u[xp, t],{xp, 3}]*Cot[\[Pi](x - xp)/(2*L)], {xp, -L, x, L}, Method -> {"PrincipalValue"}] == 0,
u[-L, t] == u[L, t], u[x, 0] == 0.1*Cos[\[Pi]/L*x]}, u, {x, -L, L}, {t, 0, tmax}]

which gives me

NDSolve::delpde:Delay partial differential equations are not currently supported by NDSolve"

The warning is understandable because the function u[xp, t] is still unknow when NIntegrate is evaluated. Note that we should use PrincipalValue here in NIntegrate because there is a singularity at $x=xp$, which has been specified in the integration range.

Comments

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